A  METEOROLOGICAL  TREATISE 

ON  THE 

o  n  £.  £_  /  c 

Circulation  and  Radiation 

IN  THE  ATMOSPHERES  OF 

THE  EARTH  AND 

OF  THE  SUN 

BY 

FRANK  H.  BIGELOW,  M.A.,  L.H.D. 

Professor  of  Meteorology  in  the  U.  S.  Weather  Bureau,  1891-1910,  and  in 
the  Argentine  Meteorological  Office  since  1910 


FIRST  EDITION 
FIRST  THOUSAND 


NEW  YORK 

JOHN  WILEY  &  SONS,  INC. 

LONDON:     CHAPMAN  &  HALL,    LIMITED 
1915 


Copyright,  1915,  by 
FRANK    H.   BIGELOW 


PUBLISHERS  PRINTING  COMPANY 
207-217  West  Twenty-fifth  Street,  New  York 


INTRODUCTION 

METEOROLOGY  as  a  science  has  failed  to  make  progress 
toward  definite  results  for  one  fundamental  reason.  In  a  non- 
adiabatic  atmosphere  the  terms  of  the  general  equations  of 
motion,  as  computed  from  the  ordinary  prescribed  formulas  of 
thermodynamics,  do  not  balance  as  required.  There  are  two 
errors  in  the  discussion:  (1)  There  is  a  mixture  of  the  non- 
adiabatic  and  the  adiabatic  systems,  and  (2)  the  important 
radiation  terms  have  been  omitted  from  the  general  equations. 
More  specifically,  for  the  Boyle-Gay  Lussac  Law,  P  =  p  T  Ry 
to  be  satisfied,  at  every  point,  it  has  been  customary  to  borrow 
R  =  gas  constant  from  the  adiabatic  system,  and  apply  it  in  the 
non-adiabatic  atmosphere.  For  example,  three  well  known 
treatments  follow: 

Bigelow.  v.  Bjerknes.  Margules. 

£W*T       Pl=(Tl\T^  fi./ 

Tj  P,       \TJ  PQ~  \T 


P*      \T0  PQ      \TJ  Po       \TU 

~ 


=  Constant        RI  =  R0  =  Con. 


Margules'  system  is  adiabatic,  v.  Bjerknes'  is  partly  adiabatic 
and  partly  non-adiabatic,  Bigelow's  is  strictly  non-adiabatic. 

Ta-T0  ,_Cp. 

TX-TO     "  Cu 

Now  it  is  true  that  each  system  satisfies  P  =  p  T  R,  but  the 
individual  values  of  PI,  7\,  pi,  RI,  are  very  different  in  the  three 
systems  for  the  same  initial  values  P0,  T0,  PO,  RO,  and  on  applying 
them  to  practical  observations  the  systems  that  are  not  strictly 
non-adiabatic  break  down  as  regards  computed  and  observed 
values  which  should  be  in  agreement.  We  can  easily  see  the 

iii 


489536 


IV  INTRODUCTION 

separate  consequences  by  the  following  equations  that  are  readily 
demonstrated: 

Adiabatic:     g  fa  -  Zo)  =  -  Cpa(Ta  -  To)  =  -  Pa~    °- 


g  fa   —   ZQ)    =    —  ^1  Cpa  (Ti   —    TO)    =    —   -  * 

Pad 

Non-adiabatic:     g  fa  -  z0)  =  -  HI  CplQ  (7\  -  T0)  -  HI  (Cpa  - 


g  (Zl  -ZQ)  =  -     -^      -  K  (?i2  -  <?o2)  -  (Qi  -  Co). 


Gravity  =  Pressure  -f-  Circulation  +  Radiation 
term.  term.  term.  term. 

In  the  adiabatic  system  by  definition  Ra  and  Cpa  are  constant, 
and  there  are  no  circulation  and  no  radiation.  In  the  non- 
adiabatic  system  there  are  both  circulation  and  radiation,  but 
these  depend  upon  the  departure  of  the  specific  heat  from  the 
adiabatic  value 

ni  (Cpa  ~  Cpio)  (7\  -  To)  =  K  (?2i  -  ?2o)  +  (Qi  -  <2o)  • 

Hence,  if  R  is  constant  it  is  a  contradiction  in  terms  to  discuss 
problems  of  circulation  and  radiation  in  the  adiabatic  or  partially 
adiabatic  systems,  as  has  been  universally  the  method. 
The  facts  of  observation,  furthermore,  conform  to 

r>  p 

-  ni  Cpio  (Ti  -  To)  =  --         —  > 


and  this  implies  that  circulation  and  radiation  are  required  to 
make  up  the  deficiency  in  respect  to  the  gravity  term  between 
two  strata  z\  and  ZQ.  Finally,  the  radiation  term  (Qi  —  Q0)  is 
usually  very  much  larger  than  the  kinetic  energy  term  ^  (qi2  — 
qf)y  but  it  has  never  been  incorporated  in  the  meteorological 
equations. 

For  these  reasons  the  author  has  spent  much  time  while  in 
the  United  States  Weather  Bureau,  and  especially  while  in  the 


INTRODUCTION  V 

Argentine  Meteorological  Office,  1910  to  date,  in  devising  a 
simple  adjustment  of  the  thermodynamic  adiabatic  equations, 
found  in  all  treatises,  to  an  exact  and  practical  form  of  computa- 
tion which  will  adapt  them  to  the  non-adiabatic  system  prevail- 
ing in  the  atmospheres  of  the  earth  and  of  the  sun.  The  following 
Treatise  sets  forth  this  new  method  of  discussing  the  meteorologi- 
cal problems,  with  sufficient  detail  to  enable  the  reader  to  utilize 
the  formulas  in  practical  computations.  It  contains  the  solution 
of  a  number  of  problems  that  have  heretofore  been  intractable 
along  the  old  lines  of  procedure: 

1.  The  diurnal  convection  and  the  semi-diurnal  barometric 
waves,  with  the  radiation. 

2.  The  pressures  and  temperatures  in  cyclones  and  anti- 
cyclones, with  the  circulation  and  radiation. 

3.  The  thermodynamics  of    the  atmosphere   from   balloon 
ascensions  to  great  altitudes. 

4.  The  thermodynamics  of  the  general  circulation. 

5.  The  distribution  of  the  radiation  in  all  latitudes  and  alti- 
tudes to  20,000  meters. 

6.  The  "solar  constant"  of  radiation  and  the  conflicting 
results  from  pyrheliometers  and  bolometers. 

7.  The  discrepancy  in  the  absolute  coefficient  of  electrical 
conduction  as  derived  from  the  several  apparatus  for  dissipation, 
and  for  the  number  and  velocity  of  the  ions. 

8.  The  diurnal  magnetic  variations  in  the  lower  strata  of  the 
atmosphere. 

9.  The  non-periodic  magnetic  variations  in  their  relation  to 
the  solar  radiation. 

10.  The  magnetization  and  electrical  terms  in  the  sun  at 
very  high  temperatures. 

I  wish  to  express  my  appreciation  and  gratitude  to  my  friend 
and  colleague,  Dr.  Walter  G.  Davis,  Director  of  the  Argentine 
Meteorological  Office,  for  his  courteous  co-operation  in  this  work, 
and  in  memory  of  our  good  old  Cordoba  days. 

FRANK  H.  BIGELOW. 

CORDOBA,  ARGENTINA. 
September,  1915. 


TABLE  OF  CONTENTS 

CHAPTER   I 

PAGE 

METEOROLOGICAL  CONSTANTS  AND  ELEMENTARY  FORMULAS  ....  1 

The  Status  of  Meteorology 1 

The  Constants  and  Formulas  of  Static  Meteorology  9 

Three  Series  of  Constants  in  Three  Systems  of  Units 12 

Variations  from  the  Standard  P0,  Po,  RO,  T0t  for  the  Same  Point  and 

for  Different  Points  on  the  Same  Vertical  Line 16 

The  Acceleration  of  Gravitation  18 

The  Density  of  the  Atmosphere  as  a  Mixture  of  Several  Constituent 

Gases .20 

The  General  Formulas  for  the  Mixture  of  Gases 23 

The  Inner  Kinetic  Energies,  Work  and  Heat 26 

The  Fundamental  Laws  of  Physics 28 

The  Kinetic  Theory  of  Gases  for  the  Atmosphere 30 

The  Temperatures  and  Temperature  Gradients  Observed  at  Different 

Elevations  in  the  Free  Air 32 

The  Temperature  Gradient  in  a  Plateau 34 

The  Integral  Mean  Temperature  and  Gradient 36 

The  Virtual  Temperature  37 

/d  T 

-jr 37 

The  General  Barometric  Formula 39 

Corrections  to  the  Barometer 41 

Examples  of  the  Barometric  Reduction  Tables     . 46 

CHAPTER   II 

THERMODYNAMIC  METEOROLOGY 50 

General  Formulas  for  the   Computation  of  P,  p,  R,  from  the  Ob- 
served Temperatures  T  in  a  Free  Non-Adiabatic  Atmosphere   .  50 

The  Adiabatic  Equations 52 

The  Working  Non-Adiabatic  Equations 53 

The  Variable  Values  of  n  =  ^ 57 

The  Differentiation  of  (172) 58 

The  Two  Laws  of  Thermodynamics 64 

The  First  Law  of  Thermodynamics 65 

Fundamental  Equations  and  Definitions 68 

The  Second  Law  of  Thermodynamics 70 

The  Reversible  Process 71 

The  Irreversible  Process 71 

Carnot's  Cyclic  Process 72 

vii 


Vlll  CONTENTS 

PAGE 
THERMODYNAMIC  METEOROLOGY — Continued 

Cyclic  Process  for  Vapors  at  Maximum  Pressure 73 

The  Second  Form  of  the  Equations  for  Latent  Heat   .    • 75 

Specific  Heats       76 

Examples  of  the  Thermodynamic  Data 77 

Application  of  the  Thermodynamic  Formulas  to  the  Non-Adiabatic 

Atmosphere 80 

Application  of  the  Thermodynamic  Formulas  to  Various  Meteoro- 
logical Problems 90 

The  Isothermal  Region       91 

The  Diurnal  Convection  and  the  Semi-Diurnal  Waves  in  the  Lower 

Strata 98 

The  Thermodynamic  Structure  of  Cyclones  and  Anticyclones  .     .     .  104 

The  Planetary  Circulation  and  Radiation 113 

The    Thermodynamic   Tables   of   the    Planetary    Circulation    and 

Radiation       118 

CHAPTER   III 

THE  THERMODYNAMICS  OF  THE  ATMOSPHERE 135 

The  Co-ordinate  Axes 135 

The  Co-ordinate  Velocities  and  Accelerations 136 

The  Constituents  of  the  Force  in  any  Direction 136 

The  Forces  of  Inertia,  Expansion,  and  Contraction 137 

The  Forces  of  Rotation 139 

The  Pressure  Gradients 143 

The  Potential  Gradient 144 

The  Equations  of  Continuity 145 

The  Operator  V2,  and  the  Total  Differential  j-{ 146 

Summary  of  the  Equations  of  Motion  in  Rectangular,  Cylindrical, 

and  Polar  Co-ordinates 147 

The  Equations  of  Motion  for  the  Rotating  Earth  in  Cylindrical  Co- 
ordinates     148 

in  Polar  Co-ordinates 149 

Connection  between  the  peneral  Equations  of  Motion  and  the  Ther- 
mal Equations  of  Energy 151 

The  Equations  for  the  Work  of  Circulation 153 

CdP 
The  Evaluation  of  the  Term  —  /  — 154 

Numerical  Check  on  the  Two  Systems  of  Formulas 157 

Numerical  Evaluations  of  the  Pressure  Gradient 157 

Evaluation  of  the  Ratios  -j-^  and  -j— 159 

dB          dx 

To  Find  the  Difference  of  Pressure  (Bi  —  B)  at  the  Distance  DI  that 

will  just  Balance  the  Eastward  Velocity  v 159 

The  Angular  Velocity  of  the  Earth's  Rotation,  a>3 160 

The  Linear  Absolute  and  Relative  Velocities 161 


CONTENTS  ix 

PAGE 

THE  THERMODYNAMICS  OF  THE  ATMOSPHERE — Continued 

Evaluation  of  the  Barometric  Gradients 162 

Application  of  the  General  Equations  of  Motion  to  the  Local  Circula- 
tions in  the  Earth's  Atmosphere 163 

Discussion  of  the  Cylindrical  Equations  of  Motion 164 

Ferrel's  Local  Cyclone 164 

The  German  Local  Cyclone 166 

The  General  Equation  of  Cylindrical  Vortices 168 

The  Angular  Velocity 171 

The  Total  Pressure 171 

Application  of  the  Vortex  Formulas  to  the  Funnel-Shaped  Tube     .  172 

to  the  Dumb-bell-Shaped  Tube 172 

The  Total  Pressure 173 

The  Relation  Between  Successive  Vortex  Tubes 173 

The  Second  Form  of  the  Cylindrical  Equations  of  Motion  in  Terms 

of  the  Current  Function  ^ 175 

The  Funnel-Shaped  Vortex 175 

The  Dumb-bell-Shaped  Vortex 177 

The  Deflecting  Force 178 

The  Force  of  Friction 179 

The  Transformation  of  Energy  and  the  Circulation  of  the  Atmosphere  180 

Case  I.  The  Change  of  Position  of  the  Layers 181 

The  Evaluation  of  /  T  dm  in  Linear  Vertical  Temperature  Changes  183 

Case  II.  Effect  of  an  Adiabatic  Expansion  or  Contraction  in  a  Non- 

Adiabatic  Temperature  Gradient 184 

Case  III.  The  Overturn  of  Deep  Strata  in  the  Column 185 

Case  IV.  The  Transformation  of  Two  Masses  of  Different  Tempera- 
tures on  the  Same  Level  into  a  State  of  Equilibrium     ....  187 
Case  V.  Local  Changes  between  Two  Strata  of  Different  Tempera- 
tures      188 

The  General  Circulation  on  a  Hemisphere  of  the  Earth's  Atmosphere  189 
Three  Cases  of  the  Slope  of  the  Temperature  Gradients  and  the  Re- 
sulting Velocity  of  the  East  and  West  Circulations 191 

CHAPTER   IV 

EXAMPLES    OF    THE    CONSTRUCTION    OF    VORTICES    IN    THE    EARTH'S 

ATMOSPHERE 196 

The  Cottage  City  Water  Spout,  August  19,  1896 199 

The  St.  Louis  Tornado,  May  27, 1896 202 

The  De  Witte  Typhoon,  August  1-3,  1901 205 

The  Ocean  and  Land  Cyclones 207 

The  Ocean  Cyclone,  October  11,  1905 209 

The  Composition  of  Vortices 212 

The  Reversed  Dumb-bell  Vortex 213 

Historical  Review  of  the  Three  Leading  Theories  Regarding  the 
Physical  Causes  of  Cyclones  and  Anticyclones  in  the  Earth's 
Atmosphere 216 


X  CONTENTS 

PAGE 

EXAMPLES   OF   THE    CONSTRUCTION    OF    VORTICES    IN   THE    EARTH'S 
ATMOSPHERE — Continued 

(1)  Ferrel's  Warm- and  Cold-Center  Cyclones 216 

(2)  Hann's 'Dynamic  Cyclonic  Whirls .     .  216 

(3)  Bigelow's  Asymmetric  Cyclones  and  Anticyclones 216 

The  General  and  the  Local  Components 220 

The  Normal  and  the  Local  Velocities  in  Storms 221 

The  Normal  and  the  Local  Isobars  in  Cyclones  and  in  Anticyclones  .  229 

The  Graphic  Construction  of  Resultants 232 

The  Normal  and  Local  Isotherms  in  Cyclones  and  in  Anticyclones  .    .  235 
The   Normal   and   Local  Velocity  Vectors  in  Cyclones  and  Anti- 
cyclones       240 

The  Land  Cyclone 243 

Recapitulation  of  the  Formulas  for  the  Dumb-bell-Shaped  Vortex  .  250 

The  Meaning  of  the  Tangential  Angle  i 251 

Example  of  the  Evaluation  of  the  Terms  in  the  Equations  of  Motion 

for  a  Cyclone 254 


CHAPTER  V 

RADIATION,    IONIZATION,  AND    MAGNETIC  VECTORS    IN    THE  EARTH'S 

ATMOSPHERE 261 

The  Determination  of  the  Intensity  of  the  Solar  Radiation  by  Obser- 
vations with  the  Pyrheliometer  and  the  Bolometer 262 

The  Pyrheliometer • 263 

Example  of  the  Practical  Observations  with  the  Pyrheliometer  at 

La  Quiaca,  Argentina,  Sept.  22,  1912 264 

The  Bouguer  Formula  of  Depletion 267 

The  Depletion  of  the  Incoming  Radiation  from  a  Maximum  Value 
on  the  Cirrus  Levels,  as  Determined  by  Observations  at  Different 

Heights 274 

The  Relative  Efficiency  of  1  Gram  of  Aqueous  Vapor  .per  Cubic 

Meter  in  Absorbing  the  Incoming  Radiation 276 

The  Bolometer  and  Its  Energy  Spectrum  of  Radiation 277 

Evaluation  of  the  Wien-Planck  Formula  of  Radiation 278 

The  Values  of  -r4f  in  the  Radiation  Formula 290 

dz 

The  Measures  of  the  lonization  of  the  Atmosphere 292 

Notation  and  Elementary  Relations 293 

Electrostatic  Relations  per  Unit  Length 295 

Conduction  lonization  Currents 296 

Coefficient  of  Electrical  Dissipation  of  the  Atmosphere  ^     .     .     .     .  297 

Elster  and  Geitel  Dissipation  Apparatus 297 

Ebert  Ion  Counter 298 

The  Formulas  for  the  Velocity  of  the  Ions 299 

Ebert  Velocity  Apparatus 300 

Gerdian  Apparatus  for  the  Number  and  Velocity  of  the  Ions    .     .     .  302 


CONTENTS  XI 

PAGE 

RADIATION,    IONIZATION,    AND    MAGNETIC    VECTORS    IN    THE    EARTH'S 

ATMOSPHERE — Continued 
The  Cause  of  the  Discrepancy  in  the  Values  of  the  Conductivity  of  the 

Atmosphere  as  Determined  by  Two  Methods 304 

The  Atmospheric  Electric  Potential 309 

CHAPTER  VI 

TERRESTRIAL  AND  SOLAR  RELATIONS 312 

The  Five  Types  of  the  Diurnal  Convection  in  the  Earth's  Atmosphere  312 
The    Diurnal    Variations    of    the   Meteorological,    Electrical,   and 

Magnetic  Elements 317 

The  Diurnal  Variations  of  the  Terrestrial  Magnetic  Fields     .     .     .  323 

The  Aperiodic  Magnetic  Vectors  along  the  Meridians 329 

The  Synchronous  Annual  Variations  of  the  Solar  and  the  Terres- 
trial Elements     .     . 335 

The  Aqueous  Vapor  in  the  Atmosphere 342 

The  Laws  of  the  Evaporation  of  Water  from  Lakes,  Pans,  and  Soils 

with  Plants 343 

The  Polarization  of  Sunlight  in  the  Atmosphere 348 

Solar  Physics 352 

The  Spherical  Astronomy  of  the  Sun 356 

The  Magnetic  Fields  of  the  Earth  and  the  Sun 360 

Conclusion 363 

CHAPTER  VII 

EXTENSION  OF  THE  THERMODYNAMIC  COMPUTATIONS  TO  THE  TOP  OF  THE 

ATMOSPHERE 364 

Remarks  on  the  Bouguer  Formula 364 

I.  First  Distribution  of  Temperature 365 

Illustrations  of  the  Use  of  Erroneous  Densities    . 372 

The  Thermodynamic  Terms 374 

The  Constituents  of  the  Solar  and  the  Terrestrial  Radiations  in  the 

Earth's  Atmosphere 375 

The  Total,  Transmitted  and  Absorbed  Amounts  of  the  Solar  and 

the  Terrestrial  Radiations     .     . 377 

II.  Second  Distribution  of  Temperature 380 

Summary  of  the  Computations  for  Twenty-one  Balloon  Ascensions  .  384 

The  Effective  Energy  of  Radiation  and  the  Solar  Constant    ....  388 

The  Second  Method  of  Discussing  Pyrheliometric  Data       ....  395 

The  Annual  Mean  Variations 401 

The  General  Summary 403 

General  Remarks 404 

The  Constants  and  Coefficients  of  Dry  Air  in  the  Kinetic  Theory  of 

Gases  throughout  the  Atmosphere 406 

Summary  of  the  Dimensions  with  Special  Reference  to  the  Equiv- 
alents      414 

General  Problems  in  Atmospheric  and  Solar  Physics        417 


A  METEOROLOGICAL  TREATISE  ON  THE 

CIRCULATION   AND    RADIATION   IN 

THE    ATMOSPHERES    OF    THE 

EARTH  AND  OF  THE  SUN 

CHAPTER  I 
Meteorological  Constants  and  Elementary  Formulae 

The  Status  of  Meteorology 

MODERN  meteorology  may  be  said  to  have  begun  its  "scientific 
development  about  the  year  1870.  Since  that  time  an  enormous 
mass  of  observations  has  been  made,  covering  every  branch  of  the 
subject,  but  the  classification  of  the  data  in  any  systematic  form 
has  been  singularly  inadequate.  This  defective  progress  may  be 
attributed  chiefly  to  two  causes,  the  first,  the  practical  units  in 
which  the  instrumental  readings  are  commonly  made,  and  the 
second,  the  usual  misapplication  of  the  Boyle-Gay  Lussac  Law 
in  the  atmosphere,  whereby  it  is  assumed  that  the  gas  coefficient 
is  constant  and  the  conditions  strictly  adiabatic.  Thus,  for 
the  pressure  P0,  thejiensity  pQ,  the  temperature  T0,  and  the  gas 
coefficient  ^0,  the  law  is, 

(1)  Po  =  PO  #o  To, 

and  it  must  be  satisfied  at  every  point,  but  in  passing  from  one 
point  to  another,  whereby  PI  p\  TI  change  values,  the  observa- 
tions made  in  balloon  and  kite  ascensions  are  not  verified  without 
making  R  change  with  the  height  above  the  sea-level.  This 
variation  of  R  carries  with  it  a  variability  of  the  specific  heat  by 
the  formula, 

(2)  Cp  =  -L-Ri 

where  the  ratio  of  the  specific  heats  is  k  =  Cp/Cv,  and  thence 
the  changes  from  one  level  to  another  follow,  for  the  free  heat 
02 1~  Co),  the  entropy  (Si  -  50),  the  work  (Wi  -  W0),  the  inner 
[1] 


2      *  '    '  -'KLETEORdtbGlCAL  CONSTANTS  AND  FORMULAS 

energy  (Ui  —  Uo),  the  radiation  energy  (Ki  —  K0),  and  the 
coefficient  c  and  exponent  a  in  the  radiation  function, 

(3)  K  =  c  T°. 

It  can  be  proved,  furthermore,  that  unless  Cp  varies  from  the 
adiabatic  specific  heat  Cpa  for  Ra  constant,  there  can  be  no 
circulation  and  no  radiation  of  heat  in  the  atmosphere,  such  as 
actually  exists.  It  is  the  purpose  of  this  Treatise  to  develop 
the  working  formulae  on  a  systematic  plan,  so  that  they  shall  be 
adapted  to  the  computation  of  the  necessary  data  throughout 
the  atmosphere,  as  in  the  diurnal  convection,  the  local  and 
cyclonic  circulation,  and  in  the  general  planetary  circulation,  by 
means  of  which  numerous  problems  may  be  studied  without 
an  undue  amount  of  speculation. 

The  difficulty  regarding  the  type  of  the  observations  available 
is  that  the  records  are  not  suitable  for  use  in  the  formulae  without 
special  reductions,  and  that  a  number  of  different  sets  of  units 
are  employed.  There  are  three  systems  of  units  in  more  or  less 
consistent  use,  (l)  the  meter-kilogram-second-Centigrade  degree 
system  (M.  K.  S.  C°),  (2)  the  centimeter-gram-second-Centi- 
grade  degree  system  (C.  G.  S.  C°),  (3)  the  foot-pound-second- 
Fahrenheit  degree  system  (F.  P.  S.  F°).  The  pressure,  Bn, 
for  the  fundamental  formulae  is  recorded  in  barometric  milli- 
meters of  mercury,  or  in  inches  of  mercury,  instead  of  in  force 
units,  employing  the  acceleration  of  gravity, 

(4)  g^=  Q.8060  (1  -  0.00260  cos  20)  (l  -   y), 

where  9.8060  in  meters  is  the  acceleration  per  second  in  latitude 
y  =  45°?  and  at  sea-level  z  =  0,  for  R  =  6370191  meters,  the 
radius  of  the  earth.  The  pressure  of  one  atmosphere  in  units 
of  force  is, 

(5)  P0  =  gOPmB0, 

where  pm  =  13595.8  kilograms  per  cubic  meter,  the  density  of 
mercury.  Similarly, 

(6)  Po  =  go  PO  Jo, 

where  p0  =  1.29305  density  of  air  in  the  same  units,  and  /0  = 


STATUS   OF  METEOROLOGY  3 

7991.04  the  height  of  the  homogeneous  atmosphere.  For  Bn  = 
0.760  meter,  the  pressure  P0  =  101323.5  kilograms  per  square 
meter.  Again,  in  units  of  mass, 

(7)  po  =  —  =  pm  Bo  =  Po  h=  10332.8. 

Finally,  in  heat  units, 

Pn 

=  24.2106 


where  the  mechanical  equivalent  of  heat  is 

(9)  A  =  426.837  kilogram  meters,  one  large  calorie. 
The  pressure  of  the  atmosphere  is,  therefore, 

Po  =  101323.5  in  units  of  force, 

po  =    10332.8  in  units  of  mass, 

pA  =    24.2106  in  units  of  heat, 
BQ  =        0.760  in  barometric  units, 

so  that  the  common  barometer  readings  must  be  wholly  trans- 
formed for  practical  computations  in  dynamic  and  thermody- 
namic  problems  which  involve  circulation  and  radiation. 

Similarly  the  (C.  G.  S.  C°)  system  and  the  (F.  P.  S.  F°) 
system  each  involves  four  other  values  for  one  atmospheric 
pressure,  and  it  often  happens  that  there  occur  crosses  between 
these  three  systems,  so  that  meteorological  literature  is  confused 
and  difficult  to  comprehend  in  any  clear  manner. 

The  temperature  is  usually  recorded  in  Centigrade  degrees, 
C°,  or  in  Fahrenheit  degrees,  F°,  but  the  computations  must  be 
executed  in  degrees  of  absolute  temperature. 

(10)  T  =  273°  +  /  (C°)  Centigrade, 

(11)  T  =  459.°4  +  /  (F°)  Fahrenheit, 

so  that  another  series  of  transformations  is  required.  It  would 
improve  matters  greatly  to  mark  barometers  in  terms  of  PO  and 
thermometers  in  degrees  T,  and  reconstruct  the  entire  series  of 
working  tables.  The  relations  between  the  barometric  and  the 
force  pressures  are  as  follows : 


4 


METEOROLOGICAL   CONSTANTS   AND   FORMULAS 


TABLE    1 
1  Scale  division  =  0.75  mm.  A  5  =  100.  AP 


A5 

in  millimeters 

AP 
units  of  force 

Unit  Equivalents 

100. 

13332. 

10. 

1333.2 

1.0  mm.  A  B  =  133.32  A  P 

1. 

133.32 

0.1 

13.332 

0.75  mm.  A  B  =  100.  A  P 

75. 

10000. 

7.5 

1000. 

0.0075mm.  A  B  =  1.0  A  P 

0.75 

100. 

0.075 

10. 

If  the  divisions  on  the  barometer  scale  are  made  0.75  millimeter, 
the  scale  distance  is  100  units  of  force  in  the  (M.  K.  S.  C°)  system, 
so  that  the  readings  of  the  barometric  pressure  are  immediately 
available  for  all  forms  of  dynamic  and  thermodynamic  meteor- 
ology. This  scale  is  equally  valuable  for  public  purposes  and 
synoptic  chart  construction. 

If  the  absolute  temperatures  (J^,  7\)  are  observed  on  two 
levels  of  the  atmosphere  at  the  heights  (z0,  £1),  respectively,  the 
formulas  permit  the  corresponding  values  to  be  computed  for 
(Po,  Pi),  (PO,  PI),  CRo,  Ri),  and  (Cpo,  Cpi).  It  can  be  shown  that 


T,  fl, 


FIG.  1.    The  temperature,  velocity,  and  heat  in  the  stratum  (zl  —  20) 

if  Cpa  is  the  specific  heat  for  the  adiabatic  Ra,  and  Cpw  is  the 
mean  specific  heat  for  the  stratum  (z0,  Zi),  the  circulation  and 
the  radiation  are  given  by  the  formula, 


(12)     (Cpa  -  Cplt)  (Ta  - 


-  Qo), 


STATUS   OF   METEOROLOGY  5 

where  the  terms  are  related  to  the  points  as  indicated  in  Fig. 
1  for  the  stratum  (z0,  Zi). 

If  (Ta  —  To)  is  the  adiabatic  temperature  fall  in  the  vertical 
distance  (si  —  z0),  and  (Ti  —  T0)  the  actual  observed  tempera- 
ture fall,  the  ratio  between  them  is, 

,  -   (Ta  ~  TQ)  dp  Qi  -  Zp)      _0o 

=  -(T,-  To)  ""   a(Zl-z0)     -  a' 

Again,  if  (q0,  <?i)  are  the  corresponding  velocities  without  re- 
gard to  direction,  and  (Qi  -  Q0)  the  loss  of  free  heat  in  the 
stratum,  these  quantities  are  all  connected  together  by  the 
equation  (12).  This  states  that  the  kinetic  energy  of  the  circula- 
tion for  the  unit  mass  ^  (<?i2  —  q02),  together  with  the  heat 
exchange  of  the  radiation  energy  of  the  free  heat  (Qi  —  Q0), 
depends  upon  the  divergence  of  the  actual  mean  specific  heat 
Cpio  from  the  adiabatic  value, 

(14)  .     Cpa  =  993.5787  at  constant  pressure. 
The  corresponding  values  for  Cva  amd  Ra  are 

(15)  Cva  =  706.5453  at  constant  volume. 
Cpa  -  Cva  =  Ra  =  287.0334  the  gas  coefficient. 

Hence.    -~^  =  k   =   1.4062486  the  ratio  of  the  specific  heats. 

Cva 

(16)  Cpa  -  Cva  =  Cva  (k  -  1) 


fto\  fl  a 

(18)       777  -  7;  —  =  —5-  =  r  --  7  =  2.461545. 

Cpa  -  Cva  Ra  k   -    I 

The  corresponding  values  in  a  non-adiabatic  atmosphere  become, 
Cp  Cp          k 


(19) 
(20) 


Cp  -Cv    '     R 
Cv  Cv 


cp  -  Cv      R     k  -  r 

f   T    \n  —  l 

where  R  =  Ra  l^r/         as  will  be  proved. 


6  METEOROLOGICAL  CONSTANTS  AND  FORMULAS 

These  two  systems  will  be  more  fully  developed,  it  being  the 
purpose  here  merely  to  point  out  the  leading  line  of  the  construc- 
tion of  this  work.  By  the  definition  of  the  specific  heat, 

(21)     +  Cpa  (T.  -  T.)  =  -  go  (»i  -  z«), 
for  the  unit  mass,  so  that 

=  -~  =  0.0098695  C°/meter, 

(23)       a  =  -  *1-jt°  =  ^  =-12-  =  ^-  C°/meter. 
Zi  —  ZQ        tpio       nCpa        n 

From  (17),  (19),  (22),  and  (23) 


k  —  1          Ra         Ra.  do       Ra  a  Ra     T\—  TQ 

It  will  be  shown  from  the  data  of  observation  that 


(25)  C#,,  (Ta  -  To)  =  ±2—Z£. 

PlO 

Subtracting  (25)  from  (21), 

(26)  (Cpa  -  Qio)  (Ta  -  To)  =  -  ^0  (21  -  z0) ^^ 

Pio 

Equating  (12)  and  (26), 


(27)     -  go  (Zl  -**)  =  +  -  +  Y2  fe2  -  go2)  +  (Qi  -  Co), 

Pio 

and  this  is  the  general  equation  of  condition  in  the  atmosphere, 
showing  that  for  the  unit  mass  the  force  of  gravitation  is  balanced 
by  the  change  of  pressure,  the  kinetic  energy  of  the  circulation, 
and  the  radiating  heat. 

Now,  returning  to  the  registration  of  the  fundamental  quan- 
tities, in  addition  to  the  temperature  T,  and  the  pressure  P,  there 
is  also  the  velocity  q.  While  the  sum  of  the  energy  of  the  circula- 
tion and  the  radiation  can  be  computed  through  the  difference 
of  the  specific  heats  (Cpa  —  CpiQ),  there  is  no  way  to  separate 
the  circulation  from  the  radiation  except  through  the  direct 
observation  of  the  velocity.  The  radiation  must  be  computed 


STATUS    OF   METEOROLOGY 


indirectly  through  the  gravitation,  pressure,  and  circulation 
terms  taken  together  in  an  inequality.  As  these  three  terms 
seldom  balance  in  the  free  atmosphere,  which  is  continually  ex- 
changing heat  at  every  point,  it  is  evident  that  the  adiabatic 
conditions,  in  connection  with  the  general  equation  of  motion, 
are  not  capable  of  giving  a  complete  solution  of  any  of  the  im- 
portant thermodynamic  problems  of  the  atmosphere.  The 
literature  of  meteorology  is  defective  in  this  respect.  It  should 
be  noted  that  the  point  of  departure  for  this  treatment  of  the 
problem  consists  in  making  R  and  Cp  variable,  as  previously 
stated. 

The  velocity  vector  q  («,  z>,  w,  a,  0)  requires  special  con- 
sideration as  to  the  axes  of  co-ordinates  and  the  angular  direc- 


Radius  of  the  Earth 


+  Z  —  direction  outward 
•+W  —  velocity  upward 


(X.Y.Z.) 
(u.v.w.) 


West 


Parallel  of  latitude 
f  y  =  direction  East 


+V    +V  =  velocity  East 


FIG.  2.     The  rectangular  co-ordinate  axes  with  component  velocities  and 

angles 

tions.  There  is  great  confusion  in  meteorology  in  the  manner  of 
recording  the  motions  of  the  atmosphere.  The  popular  use  of 
the  compass  points  giving  the  directions  from  which  the  wind 
blows  in  the  azimuth  rotation  N,  E,  S,  W,  is  entirely  inapplicable 


8  METEOROLOGICAL   CONSTANTS   AND   FORMULAS 

in  meteorological  computations.  This  system  should  be  reversed 
in  two  respects,  (1)  The  vector  direction  is  that  toward  which 
the  air  moves,  instead  of  that  from  which  it  blows,  making  an 
azimuth  difference  of  180°;  (2)  the  proper  co-ordinate  axes  make 
the  azimuth  rotation  (S,  E,  N,  W).  The  result  of  these  two 
changes  is  effected  by  the  formula. 

(28)  0  =  360°  -  A, 

where  A  is  the  azimuth  in  degrees  from  the  north  through  the 
east,  and  0  the  azimuth  from  the  'south  through  the  east,  the  vector 
being  changed  to  record  the  direction  "towards"  instead  of  the 
direction  "from"  which  the  wind  moves.  The  rectangular  co- 
ordinate axes  are  fixed  by  the  common  convention  of  a  right- 
handed  rotation  about  a  radius  of  the  earth  with  positive  trans- 
lation outward. 

(Meridian)  +  x  =  axis  positive  southward;  +  u  =  velocity 
south. 

(Parallel)  +  y  =  axis  positive  eastward;  +  v  =  velocity 
east. 

(Radius)  +  z  =  axis  positive  outward;  -f  w  =  vertical 
velocity. 

(29)  Horizontal  Plane,   s  =  (x2  +  y2)*.     «  =  (u2  +  »*)*. 

(30)  Vertical  Plane.        r  =  (x2  +  y*  +  zrf.    q  =  (u2  +  v*  +  w2)1. 

(31)  Azimuth  angle  (S,  E,  N,  W),      tan  0  =  --  =  — . 

z         w 

(32)  Vertical  angle  (above  horizon),  tana  =  --  =  — . 

The  same  co-ordinate  relations  should  be  employed  in 
terrestrial  magnetism,  atmospheric  electricity,  and  vector  physics 
generally. 

Besides  recording  the  direction  of  motion  very  inconveniently, 
the  velocities  themselves  usually  require  a  series  of  transforma- 
tions to  reduce  than  to  practical  dynamics.  The  anemometers 
are  commonly  graduated  in  kilometers  per  hour,  or  in  miles  per 
hour,  but  they  should  be  graduated  in  meters  per  second  for 


CONSTANTS  AND   FORMULAS  9 

the  (M.  K.  S.)  system,  and  in  feet  per  second  for  the  (F.  P.  S.) 
system  in  order  to  conform  with  the  other  terms  of  equation 
(27).  In  the  electrical  self-registration  of  the  wind  direction  it  is 
common  to  limit  the  compass  points  to  eight  in  number,  N,  NE, 
E,  SE,  S,  SW,  W,  NW,  but  in  all  problems  requiring  accurate 
wind  deflecting  components,  as  in  studies  of  the  diurnal  convec- 
tion, it  is  necessary  to  use  at  least  16  points  of  22.5°  each  in  order 
to  compute  the  observed,  resultant,  and  deflecting  vectors. 
Finally,  it  would  be  much  better  to  record  the  azimuth  /3  (S,  E, 
N,  W)  in  degrees  of  arc,  as  can  be  readily  done  by  a  good  me- 
chanical device.  The  vertical  angle  a  must  be  computed  instead 
of  observed,  because  it  is  small  except  in  tornadoes  and  cannot 
be  ordinarily  measured  mechanically.  It  is  very  important 
to  record  the  wind  vector  (q,  0,  a)  in  the  system  thus  described, 
in  order  to  facilitate  all  studies  in  the  higher  problems.  It 
may  be  noted  that  .there  is  evidence  to  show  that  the  wind 
velocity  recorded  by  the  anemometer  is  on  a  scale  about  20 
per  cent,  greater  in  the  United  States  than  in  Europe.  This 
subject  should  be  fully  tested  as  soon  as  possible.  It  is  also 
known  that  the  ordinary  anemometer  registers  excessive  velocities 
as  compared  with  a  force  dynamometer,  such  that  the  recorded 
value  40  means  33,  60  means  48,  80  means  62,  thus  introducing 
great  errors  in  the  dynamic  equations  unless  corrected. 

The  Constants  and  Formulas  of  Static  Meteorology 

Meteorology  distributes  itself  into  three  parts  in  accordance 
with  the  requirements  of  equation  (27).  Static  meteorology 
develops, 

(33)  -  g(*  -  2o)  = 


and  it  is  that  which  is  generally  used  in  the  construction  of 
synoptic  weather  charts  and  the  other  elementary  problems. 
Dynamic  meteorology  develops, 

(34)        -  g  (Zl  -  Zo)  =  +  ^^  +  y,  (?1«  -  <?„'), 

Pio 


10  METEOROLOGICAL  CONSTANTS  AND  FORMULAS 

and  is  concerned  with  the  several  general  equations  of  motion 
connecting  circulation  and  pressure. 

Thermodynamic  meteorology  develops, 

(27)   -  g  (ft  -  so)  =  +  ^-=-^2  +  1A  (<?i2  -  <?o2)  +  (Qi  -  Co), 

Pio 

and  unites  the  radiation  with  the  circulation  and  the  pressure 
through  the  functions  of  work  and  inner  energy.  It  follows  that 
the  term  —  g  (z\  —  z0)  may  be  broken  up  into  three  parts : 

(36)  -  g  (z  -  ZQ)  =  -  g  (zi  -  ZQ)  -  g  (zz  -  zi)  -  g  (*»  -  22) 
where  we  have,  respectively, 

r>  7> 

(33)   —  g  (zi  —  s0)  =  H °Lthe  pressure  effect, 

Pio 

(37)  -  g  (z2  -  zi)  =  +  y2  (qi2  -  g02)  the  circulation  effect, 

(38)  -  g  fa  -  «a)  =  +  (Qi  -  Qo)  the  radiation  effect. 

Each  of  these  terms  is  effective  in  disturbing  the  normal  pressure, 
temperature,  and  density  levels,  which  would  assume  fixed 
positions  when  uninfluenced  by  the  absorption  and  the  emission 
of  solar  and  terrestrial  radiation,  the  entire  process  being  the 
means  of  continually  returning  to  normal  equilibrium. 

In  order  to  derive  the  constants  and  the  formulas  for  static 
meteorology,  the  formulas  (25)  and  (33)  are  united  to  form, 

(39)  -  Pl  ~J°  =  fto  fe  -  z0)  =  -  Qio  (T,  -  To), 

where  the  mean  gravity  and  the  mean  specific  heat  between 
the  two  vertical  points,  z\  and  z0l  are  to  be  used.  Then', 

(40)  -  (Pi.-  Po)  =  £10  Pio  fe  -Zo)  =  -  pioQio  (Ti  -  To). 

Since  for  a  column  on  a  base  of  unit  square  area  pio  (z\  —  ZQ)  = 
M,  the  mass  that  produces  the  pressure  —  (Pi  —  P0)  when  acted 
upon  by  the  force  of  gravity  gi0,  in  the  differential  equation,  is, 

(41)  -  J dP  =  J gdm  =  j gpdz  =  -•  JpCp  d  T. 

If  the  upper  limit  is  at  the  top  of  the  atmosphere,  and  the 
lower  limit  at  the  bottom  of  it,  on  the  sea  level  in  latitude  45°, 


CONSTANTS  AND  FORMULAS 


11 


and  for  the  temperature  T  =  273°,  (40)  reduces  to, 

(42)  Po  =  go  po  Zi  =  PO  Cpo  (273°  -  7\). 
When  the  temperature  of  reduction  is 

(43)  3Ti  -  273° 

the  last  form  in  (42)  disappears.  The  constants  of  static  meteor- 
ology conform  to  (42)  for  any  substance  whatsoever:  water, 
mercury,  dry  air,  aqueous  vapor,  or  mixtures.  If  P^  is  the 
pressure  of  one  standard  atmosphere  the  density  must  change 
Tn  an  inverse  proportion  with  the  height.  In  the  following 
notation, 


Substance 

Density 

Height 

Column 

Water 

Pw 

HUJ 

Water  column 

Mercury 

Pm 

Bo 

Barometer 

Dry  air 

Po 

/o 

Homogeneous    • 

Aqueous  vapor 

'Pz 

h 

Vapor  column 

(42)  becomes,  specifically, 

(44)  Po  =  go  Pw  hw    =     go  Pm  BQ      =      go  Po  /O  =          go  pZ  /2- 

and,  (water)       (mercury)       (dry  air)       (aqueous  vapor) 

p 

(45)  —  =  pwhw     =     pmBQ      =       pp/p_  P2/2. 

Before  evaluating  equations  (44)  and  (45),  it  is  necessary 
to  adopt  the  standard  constants  *  of  transformation  between  the 
three  fundamental  systems  (M.  K.  S.),  (C.  G.  S.),  (F.  P.  S.). 

The  equivalent  units  of  length  and  volume  are, 

(46)  1  meter   =  100  centimeters  =    3.2809  feet. 

1  meter3  =  1000000  cm3        =  35.3166  cu.  ft. 
The  standard  relation  between  volume  and  mass  is, 
1  cubic  centimeter  of  water  =  1  gram  at  the  temperature  276.9°. 

(47)  1  kilogram    =  1000  grams          =  2.20462  pounds. 
1000  kilograms  =  1000000  grams   =  2204.62  pounds. 

*The  subject  of  units  and  physical  constants  can  be  studied  in  Everett's 
"  Units  and  Physical  Constants,"  Gray's  "  Smithsonian  Physical  Tables," 
and  in  the  text-books  on  physics  generally. 


12 


METEOROLOGICAL   CONSTANTS   AND   FORMULAS 


Hence,  by  division,  the  equivalents  become, 
kilograms 


(48)1000 


, 

(meter)3  (centimeter)  3 


=  62.4237 


16.0198ki!2SI^?=  0.0160198^=     1 
m3  cm3 


(foot)3' 
pound 


Three  Series  of  Constants  in  Three  Systems  of  Units 

In  Static  Meteorology  there  are  three  series  of  constants  for 
force  units,  mass  units,  and  heat  units  in  the  three  systems  of 
units  (M.  K.  S.  C°),  (C.  G.  S.  C°),  (F.  P.  S.  F°).  These  de- 
velop from  fundamental  principles  or  definitions.  Thus,  to 
illustrate  by  pressure: 

1.  Force  pressure  =  mass  X  acceleration. 


(49) 


P0  =  pmB0go  =  po/ogo  =  M  go  =  pogo  =  PA~- 


2.  Mass  pressure  =  heat  pressure  X  mechanical   equivalent 
of  heat. 


(50) 


u      A      PA 
—  M  =  —  =  -r. 
go        A 


3.    Heat   pressure  =  force    pressure  X  heat    equivalent    of 
gravity  work. 

(51)  PA  =  AM  =  Po-  =  ApQ. 

£o 

These  transformations  apply  to  the  heat  terms  R,  Cp,  Cv,  in 
the  several  systems.     These  factors  become  in  the  several  unit 

systems : 

TABLE  2 
GRAVITY  AND  MECHANICAL  EQUIVALENTS  OF  HEAT 


Work  and  Heat  Equivalents 

(M.  K.  S.) 

(C.  G.  S.) 

(F.  P.  S.) 

go  Acceleration  of  gravity  
-£  Work  equivalent  of  heat  

—  Gravity-work  of  heat. 

9.8060 
426.837 

4185  1 

980.60 
42683.7 

41851000 

32.173 
777.93 

25028  2 

A 

A  Heat  equivalent  of  work  .  .  . 
A_  J  Heat  equivalent  of  gravity 
go  '  work  

0.002343 
0.00023894 

0.00002343 
0.000000023894 

0.0012855 
0.000039954 

A  X  (Heat  in  mechanical  units)  =  Heat  units  of  heat  =  calories. 


THREE    SYSTEMS    OF   UNITS 


13 


In  Tables  3,  4,  and  5  have  been  collected  together  the  con- 
stants in  the  three  unit  systems.  They  illustrate  practically 
the  formulas  (1),  (17),  (18),  (22),  and  (44)  in  Table  3,  (50)  in 
Table  4,  and  (51)  in  Table  5.  By  combining  these  constants  and 
formulas  a  very  large  amount  of  static  meteorology  is  derived. 

TABLE  3 
THREE  SERIES  OF  CONSTANTS  IN  THREE  SYSTEMS  OF  UNITS 

(1)     Gravitational  Force  Units 


Formulas 

S 

(M.  K.  S.  C°) 
Meter-Kg-Second 

(C.  G.  S.  C°) 
Cm.-Gram-Second* 

(F.  P.  S.  F°) 
Foot-Pound-Second 

go 
pw 
hw 
Po 

go 
pm 
Bo 
Po 

go 

PO 

It 

Po 

go 

P2 

h 

Po 

ft 

k  -1 
ft 

k-l 
1 
»-l 

Ro 
To 
po 
Po 

lo 
go 

a 
Ro 

Cpa 
Cva 

ao 

Log. 
9.8060           0.99149 
1000  .0           3  .  00000 
10.3329         1.01422 
101323.5       5.00571 

9  .  8060          0  .  99149 
13595.8        4.13340 
0  .  760            9  .  88081 
101323  .5      5  .  00571 

9.8060           0.99149 
1.29305         0.11162 
7991.04         3.90260 
101323.5       5.00571 

9.8060           0.99149 
0.80427         9.90540 
12847.6         4.10882 
101323.5       5.00571 

1.4062486     0.14806 
0.4062486     9.60879 
3.461545       0.53927 

2.461545       0.39121 

287.0334       2.45793 
273.                2.43616 
1.29305         0.11162 
101323.5       5.00571 

7991.04         3.90260 
9.8060           0.99149 

0.003663       7.56384 
287.0334       2.45793 

993.5787       2.99720 
706.5453       2.84914 

0.0098695     7.99429 

Log. 
980.60           2.99149 
1.0000           0.00000 
10332.9         3.01422 
1013235.       6.00571 

980.60          2.99149 
13  .  5958        1  .  13340 
76.0               1.88081 
1013235.       6.00571 

980.60           2.99149 
.00129305     7.11162 
799104.          5.90260 
1013235.        6.00571 

980.60           2.99149 
.00080427     6.90540 
1284760.        6.10882 
1013235.        6.00571 

1.4062486     0.14806 
0.4062486     9.60879 
3.461545       0.53927 

2.461545       0.39121 

2870334.       6.45793 
273.                2.43616 
0.00129305  7.11162 
1013235.        6.00571 

799104.          5.90260 
980.60           2.99149 

0.003663       7.56384 
2870334         6.45793 

9935787.        6.99720 
7065453.       6.84914 

.000098695  5.99429 

Log. 
32.173           1.50749 
62.4237         1.79535 
33.901           1.53021 
68085             4.83305 

32  .173          1  .  50749 
848.70          2.92875 
2.4935          0.39681 
68085            4  .  83305 

32.173           1.50749 
0.080717       8.90696 
26218.1         4.41860 
68085.           4.83305 

32  .173           1  .  50749 
0.056009       8.69974 
42249.1         4.62582 
68085.           4.83305 

1.4062486     0.14806 
0.4062486     9.60879 
3.461545       0.53927 

2.461545       0.39121 

1716.52         3.23465 
491.4             2.69144 
0.080717       8.90696 
68085             4.83305 

26218.1         4.41860 
32.173           1.50749 

0.002035       7.30856 
1716.52         3.23465 

5941.86         3.77392 
4225.34         3.62586 

0.0054146     7.73357 

Density  of  water  

Height  

(44)    Po  =  gopwhw.... 
Gravity 

Density  of  mercury  .  .  . 
Barometer  height  
(44)  Po  =  gopmBo.  .  .  . 

Height  

(44)  Po  -  go  polo  

Gravity 

Density  aqueous  vapor. 
Height  

(44)  Po  =  gop2  /2  

Cp 
Specific  heats  -^  

Ratio  ^  -  1  .  . 
C  v 

(17)  Const,  press  
(18)  Const,  vol  

The  Boyle-Gay  Lussac 
Law 

(1)  Po  =  PoRoTo  
lo  go  =  Ro  To  

(1)  and  (44)  
1 

a,  =  "Tfr 

To 
Ro  ~  lo  So  a. 

(17)  CP  =Ko  jp-j.... 
(18)  Cv  =  Ro  jri—  • 

(22)-d/-^°- 

'     d  z       Cpa 

14 


METEOROLOGICAL  CONSTANTS  AND  FORMULAS 


(2) 


TABLE  4 

Mass  or  Units  of  Weight 


Formulas 

5 

(M.  K.  S.  C°) 
Meter-Kilgm.-Sec. 

(C.  G.  S.  C°) 
Cm.-Gm.-Sec. 

(F.  P.  S.  F°) 
Ft.-Pound-Sec. 

(50)  p  =  Pa/go  

R    =  p/poTo.... 

P 
R 

CP 

Cv 

Log. 
10332.8       4.01422 
29.2713       1.46644 

101.3235      2.00571 
72.0522       1.85765 

Log. 
1033.28       3.01422 
2927.13       3.46644 

10132.35       4.00571 
7205.22       3.85765 

Log. 
2116.20         3.32556 
53.353         1.72716 

184.683        2.26643 
131.330        2.11837 

Ci>-R-^— 

Cv=Cp  -  R  

TABLE  5 

(3)     Heat  Units 


Formulas 

S 

(M.  K.  S.  C°) 

(C.  G.  S.  C°) 

(F.  P.  S.  F°) 

Work  equiv.  heat  

1 

A 

A 

PA 

RA 

CPA 
C»A 

426.837        2.63022 
0.002343       7.36978 
24.2106         1.38400 
0.068583       8.83622 
0.237406       9.37549 
0.168823       9.22743 

42683.7        4.63022 
0.00002343  5.36978 
0.024106       8.38400 
0.068583       8.83622 
0.237406       9.37549 
0.168823       9.22743 

777.93  2.89094 
0.0012855  7.10906 
2.72025  0.43461 
0.068583  8.83622 
0.237406  9.37549 
0.168823  9.22743 

Heat  equiv.  work  

(5D  PA=AP  
RA  =  AR  

C*A  =  ACp  
CvA   =  ACv...    . 

Work  and  Heat  Units 

One  large  calorie  is  the  heat  required  to  raise  1  kilogram  of 
water  from  0°  to  1°  C. 

One  small  calorie  (therm.)  is  the  heat  required  to  raise  1 
gram  of  water  from  0°  to  1°  C. 

One  British  thermal  unit  is  the  heat  required  to  raise  1  pound 
of  water  from  32°  to  33°  F. 

One  calorie  =  1000  therms  =  3.968  Br.  th.  u.  =  426.837 
kilogram  meters. 

One  therm  =  0.003968  Br.  th.  u.      (3.968  =  2.2046  X  1.8.) 

One  dyne  is  the  force  which  acting  upon  a  gram  for  one  second 
generates  a  velocity  of  one  centimeter  per  second;  it  produces 
the  C.  G.  S.  unit  of  acceleration  on  one  gram;  it  produces  the 
C.  G.  S.  unit  of  momentum  on  any  mass  per  second. 

One  erg  is  the  amount  of  work  done  by  one  dyne  working 


THREE   SYSTEMS   OF  UNITS  15 

through  the  distance  of  one  centimeter;  it  is  the  C.  G.  S.  unit  of 
energy. 

One  erg  =  1  centimeter  dyne  =  ^TTT;;  =  0.0010198  gram  cm. 


-  980.60  xUx  100  =  0-000000010198  kilogram 

meter. 

One  large  calorie  =  1  kilogram-degree  C°  water  =  426.837 

kilogram  meters 

=  426.837    X  980.60   X   1000    X    100  = 
4.1851  X  1010  ergs,  C.  G.  S. 

One  small  calorie  =  1    gram-degree   C°   water  =  426.837 

gram  meters 

=  426.837  X  980.60  X  100  =   4.1851  X 

107  ergs,  C.  G.  S. 
One  British  thermal  unit  =  1  pound-degree  F°  water 

o  280Q 
=  426.837  X  -      -=  777.93  foot-pounds. 

l.o 


Work  to  Heat 

—T.  The  mechanical  equivalent  of  heat  is  the  work  required 
by  work-friction  to  produce  the  given  heat. 

Log. 

(52)  -j  =  426.837        2.63022  kilogram  meters    (M.  K.  S.) 

=  42683.7        4.63022  gram  centimeters  (C.  G.  S.) 
=  777.93         2.89094  foot-pounds  (F.  P.  S.) 

(53)  il=  4185.1         3.62171  joules  (M.  K.  S.) 

=  41851000.    7.62171  ergs  (C.  G.  S.) 

=  25028.2       4.39843  absolute  units      (F.  P.  S.) 

Heat  to  Work 

A.  The  heat  equivalent  of  work  is  the  heat  that  is  required 
to  do  a  given  amount  of  work. 


16  METEOROLOGICAL   CONSTANTS   AND   FORMULAS 

Log. 

(54)  A  =  0.002343  7.36978  kilogram  calorie  (M.  K.  S.) 

=  0.00002343          5.36978  gram  therm  (C.  G.  S.) 

=  0.0012855  7.10906  Br.  th.  units  (F.  P.  S.) 

(55)  —  =  0.00023894          6.37829  (M.  K.  S.) 

=  0.000000023894  2.37829  (C.  G.  S.) 

=  0.000039954        5.60157  (F.  P.  S.) 


I.  FOR  THE  SAME  POINT  OR  STATION 
Variations  from  the  Standard  P0,  p0,  RQ,  TQ 

The  several  formulas  derived  from  P0  =  PO  ^o  TO  apply 
only  to  the  sea  level  on  latitude  45°,  but  the  variations,  PI,  pi,  RI, 
TI,  are  incessant  in  the  earth's  atmosphere,  and  the  formulas 
must  be  derived  for  passing  from  one  condition  to  another. 
Meteorology  divides  itself  into  two  main  branches  according  as 
R  is  taken  constant  or  variable,  and  it  is  a  principal  part  of  this 
work  to  discuss  the  formulas  when  R  is  variable  from  one  stratum 
to  another.  When  the  point  of  observation  is  on  the  sea  level  or 
on  the  lower  land  areas,  it  is  proper  to  assume  RI  =  RQ  constant, 
which  greatly  simplifies  the  computations.  If  the  variations  oc- 
cur at  the  same  place  or  station,  go  is  also  constant.  For  two 
variable  conditions  of  dry  air  at  a  given  place  we  may  write  the 
two  in  a  ratio  in  several  forms,  using  (44),  (49),  (50),  (51) : 


Hence, 


. 
PAOJ 


,__N     P_          j>_  pA      _  P    T    RQ    _   Pm  B   go   _      p  I  go 

PO  pQ  pAQ  PoToRo  Pm^ogo  PO  IQ  go 

We  shall  confine  our  attention  chiefly  to  the  force  pressures 
P,  PO  in  the  (M.  K.  S.)  system,  and  to  the  first  series  of  constants 
in  Table  3.  There  are  numerous  equivalents  which  are  easily 
derived  from  the  formulas. 


VARIATIONS   IN   THE   VERTICAL   LINE  17 

P  T         o  I         B 

(58)  The  pressure  ratio,  -^-  =  —  ^    =  —  -  =  —  . 

•LQ  PO  J-  0  PO*0  -OQ 

The  temperature  ratio  becomes, 


The  pressure  ratio  may  have  several  forms,  from  which  are 
derived  the  pressure-density  ratio, 

(60)  —  =  —  ~  =  —  (1  +  a  t).  Auxiliaries. 

p  po    1  o         po 

^  0  Pm  go   /«      i          A  D  D 

-  (1  +  a/).     Po  =  #o  Pm  go. 
po 

=  PO  ^o  (1  +  a  0-  —  =  VQ. 

Po 

=  Ro  TQ  (1  +  a  t).          P0  V0  =  R0  T0. 

T  P  D 

(61)  The  temperature  ratio,  -—=—.—. 

1  o         "  <y      P 

(62)  The  density  ratio,          —  =  ^  .  ^ 

Po        -TO      ^ 

^         -^ 

The  temperature  varies  as  the  pressure  and  inversely  as  the 
density;  the  density  varies  as  the  pressure  and  inversely  as  the 
temperature. 


II.  FOR  DIFFERENT  POINTS  ON  ANY  VERTICAL  LINE  z 
The  Variations  of  Gravity,  Density,  Temperature,  and  Pressure 

The  practical  problems  in  static  meteorology  consist  to  a 
considerable  extent  of  the  reduction  of  barometric  pressures 
from  one  elevation  to  another  along  a  radius  of  the  earth  extended, 
or  inversely  the  determination  of  the  difference  of  elevation  be- 
tween two  measured  barometric  pressures  along  the  same  vertical.  ' 
The  former  process  is  applied  in  forming  the  synchronous  charts 
of  pressure  reduced  to  the  sea  level,  or  to  any  other  adopted 
plane  which  are  used  in  public  forecast  charts  of  storm  and 
weather  conditions,  and  the  latter  to  preliminary  surveys  in 


18  METEOROLOGICAL  CONSTANTS  AND  FORMULAS 

mountain  and  plateau  regions.  The  entire  process  is,  however, 
very  complex  in  its  application  to  particular  cases,  and  it  re- 
quires much  experience  in  managing  the  details  of  the  computa- 
tions. It  will  be  necessary  to  describe  somewhat  fully  the 
several  terms  that  are  to  be  integrated  from  one  level  to  another, 
which  enter  the  final  barometric  and  hypsometric  formulas. 
Those  here  developed  are  such  as  are  found  in  the  Standard 
^Treatises,  but  another  method  of  reduction  will  be  introduced 
at  a  later  section.  The  first  problem  to  describe  is  the  gravity 
value  in  any  latitude,^,  and  at  any  height  above  the  sea  level,  z; 
the  second  considers  the  density  of  the  air  as  a  mixture  of  gases 
and  aqueous  vapor  in  varying  proportions;  the  third  is  to  de- 
termine the  temperature  gradients  in  a  vertical  direction  in  the 
free  air  and  within  the  land  masses;  and  the  fouxth  is  the  use  of 
the  barometer  as  an  instrument  of  precision,  together  with  the 
discussion  of  the  observed  heights  of  the  mercury  column.  All 
the  details  easily  found  in  good  works  on  meteorology  will  be 
very  briefly  mentioned. 

7.  The  Acceleration  of  Gravitation  J  g$z  d  z. 

From  formula  (4),  which  is  of  geodetic  origin, 
(63)  g$  =  go  (1  -  0.00260  cos  2  0),  the  latitude  variation. 

*•"  (64)  gz  =  gj  ^1  —  -~rj,  the  elevation  variation.    Hence, 
(65)  f^  d  z  =  9.8060  (1  -  0.00260  cos 

The  force  of   gravity  varies  inversely  as  the  square  of   the 
distance  from  the  center  of  the  earth. 

(66>>l£-        Rz 
(bb)  go   "  (R  +  *)•  - 


-Z0). 


(67) 


-          dz  =  (z  -  *0)  - 


ACCELERATION   OF   GRAVITATION 


19 


The  radius  of  the  earth  may  be  taken  in  the  mean  from 
Bessel's  spheroid, 

(68)  JRp  =  6370191  meters,  6.8041525  log. 
Rt    =  20899600  feet,     7.3201380  log. 

The  computed  values  of  formula  (4),  without  any  integration 
in  latitude  and  altitude,  are  given  for  a  few  selected  points  in 
Table  6. 

TABLE  6 
EVALUATION  OF  FORMULA  (4) 

gfa  =  9.8060  (1  -  0.00260  cos  2  0)  ( 1  -  ^) 


z 

90° 

80° 

70° 

60° 

50° 

45° 

40° 

30° 

20° 

10° 

0° 

Meters. 
20000 
15000 
10000 
5000 
000 

m. 
9  .  7703 
9.7857 
9.8011 
9.8165 

m. 
9.7688 
9.7842 
9.7996 
9.8150 

m. 
9.7643 
9.7797 
9.7951 
9.8105 

m. 
9.7574 
9.7728 
9.7882 
9.8036 

m. 

9  .  7489 
9  .  7643 
9.7797 
9.7951 

m. 
9  .  7444 
9.7598 
9.7752 
9.7906 

m. 
9.7399 
9.7553 
9.7707 
9.7861 

m. 
9.7314 
9.7468 
9.7622 
9.7776 

m. 
9.7245 
9.7399 
9.7553 
9.7707 

m. 
9.7200 
9  .  7354 
9.7508 
9  .  7662 

m. 
9.7185 
9.7339 
9.7493 
9  .  7647 

9.8319 

9  .  8304 

9.8259 

9.81909.8105 

9.8060 

9.8015 

9  .  7930 

9.7861 

9.7816 

9.7801 

2z 
The  variation  in  height  g0-^-  =  0.00308  meters  per  1,000 

meters,  and  0.00308  feet  per  1000  feet  for  g0  =  32.173  feefT 

There  has  been  a  discussion  as  to  the  effect  of  the  land  masses 
upon  the  action  of  gravity,  whether  the  coefficient  in  the  formula 
should  be  2.00,  as  developed  in  (66),  or  be  modified.  Ferrel 
claims  that  it  should  remain  2.00;  the  Smithsonian  Meteorologi- 
cal Tables  have  adopted  1.96;  and  the  International  Meteoro- 
logical Tables  have  taken  1.25,  which  latter  value  is  here 
adopted.  The  plateau  regions  of  North  and  South  America,  Asia, 

rr 

and  Africa  will  be  best  represented  by  1.25  -~-,  where  z  has  a  con- 
siderable value,  reckoned  Trom  the  sea  level.  In  balloon  ascen- 
sions from  the  ocean  or  from  the  low  plains  it  may  be  better  to 
increase  the  value  to  2.00,  but  this  can  be  determined  from  obser- 
vations by  means  of  mercurial  and  aneroid  barometers.  In  fact, 
the  aneroid  barometer,  when  perfectly  adjusted  as  a  mechanism, 


20  METEOROLOGICAL  CONSTANTS  AND  FORMULAS 

measures  the  local  hydrostatic  pressure  without  any  gravity 
factor.  Among  the  computations  to  be  introduced  in  a  later 
section  there  will  be  examples  of  this  action.  Admitting  the 
coefficient  1.25,  we  have 

(69)  gz  =  g0(l  —  1-25^)  =  9.806  (1  -  0.000000196s)  metric. 

=  32.173  (1  -  .0000000598  z)  English. 
=  9.806        -  0.00000192  z  metric. 
=  32.173      -  0.00000192  z  English. 

Similarly,  the  correction  for  height  can  be  applied  to  any  other 
value  of  #0  ,  as  found  on  the  lower  line  of  Table  6. 

2.  The  Density  of  the  Atmosphere  as  a  Mixture  of  Several  Con- 

stituent Gases 

For  the  practice  of  barometry  it  is  sufficient  to  take  account 
of  the  atmosphere  as  a  mixture  of  dry  air,  aqueous  vapor  and 
carbon  dioxide,  commonly  called  carbonic  acid.*  In  physical 
problems  there  are  the  gases  oxygen,  nitrogen,  hydrogen,  carbonic 
oxide,  and  traces  of  argon,  helium,  neon,  krypton.  We  shall 
summarize  the  treatment  of  several  gases  in  a  mixture  and 
the  data  of  the  kinetic  theory  of  gases  in  this  connection. 

Adopt  the  notation  for  standard  conditions  as  expressed  by 
Po  =  PO  #o  To,  or  Po  v  0=  Ro  TV 


Mixture.        Dry  Air.     AJueous      CarAb™ic 

Vapor.          Acid. 

Density  Pm0  p^  ^  P30 

Volume  vm0  vw  v20  v30 

The  general  equation  for  mixture  is, 

(70)     pmQ  Vm(>    =    pio  fllO   +   P20  ^20   +    P30  ^30- 

The  values  to  be  assigned  to  the  terms  are: 

Dry  air  Plo  =  1.29278.      v10  =  vmo  -  v2Q  -  v30. 

*  In  view  of  the  possible  existence  of  the  real  carbonic  acid  (H2  C  O3)  at 
the  low  temperature  of  the  isothermal  layer,  the  use  of  the  word  "  acid  "  instead 
of  anhydride  can  not  be  commended. 


DENSITY   OF  THE  ATMOSPHERE  21 

Aqueous  vapor  p20  =  0.622  pi0.    v2Q  =  variable  amounts. 
Carbonic  acid    pso  =  1.529  pi0.     £30  =  0.0004  vm0. 
Mixture  pm0  to  be  computed.  vm0  =  vw  +  v20  +  v30. 

Introducing  these  values  in  (70)  and  dividing  by  vmQ 

(71)  ^  =  1.29278(1-^  -  ^  +  0.622^  +  1.529^), 

\          Z>mO         VmO  Vm0  VmO/ 

=  1.29278  fl  -  0.378  —  +  0.529—  }. 
\  vm0  vm0J 

Since  the  ratio  of  the  volume  of  the  constituent  to  the  volume 
of  the  mixture  is  the  same  as  that  of  the  partial  pressure  of  the 
constituent  to  the  total  pressure  of  the  mixture,  we  have  generally 

VnO  Pn0  pn_  e_ 

vm0  "  pm0  "  pm  "  B' 

where  pn0  and  pm0  are  for  the  normal  data  (P0  TO), 
pn   and  pm  are  for  any  (P  T)  , 
e     and  B    are  the  barometric  pressures. 

Hence,  —  =  £,  and  ^  =  0.0004,  so  that, 

Z>mO         £>Q  VmO 

(73)  pm0  =  1.29278  (l  -  0.378^  +  0.00021  V 

It  is  customary  to  unite  the  terms  for  the  dry  air  and  the 
carbonic  acid  in  the  normal  density, 

(74)  po  =  1.29278  +  0.00027  =  1.29305  per  cubic  meter. 

In  (73)  e^is  to  be  taken  in  meters  of  mercury  in  the  (M.  K.  S.) 
system,  in  millimeters  in  the  (C.  G.  S.)  system,  and  in  feet  in  the 
(F.  P.  S.)  system.  Since  e0  varies  incessantly  in  the  atmosphere 
no  fixed  value  can  be  assigned  to  it  on  any  level.  The  reduction 
from  the  normal  (PQ  pm0  T0)  to  any  other  condition  (P  pm  T)  on 
the  same  level  z0  is  given  by  (60),  substituting  pm  for  /o,  and 
Pmo  for  /°o, 

p    'T'  p     'T1  /  p 

(75)  pm    =       -          PmQ   =   ~  P™Q       1    ~   0.378       - 


p 


22  METEOROLOGICAL   CONSTANTS   AND   FORMULAS 

B  1  1 

(76)  pm  =  -5-  pm0 


6/)(l  +  0.378) 


Since  we  retain  the  density  p0  =  1.29305  for  the  value  of 
pm0,  with  dry  air  at  normal  pressure  in  this  equation,  the 
corresponding  barometric  pressure  in  the  fraction  expressing 
the  partial  pressure  of  aqueous  vapor  must  be  B0-,  butj^may 
be  any  pressure  whatever,  according  to  the  dryness  of  the  air. 

Cz  e 

The  Integral  Mean  /  0.378  -=-. 

•'so  £>o 

The  pressure  of  the  aqueous  vapor  decreases  from  the  ground 
upward  in  a  geometric  ratio,  which  is  expressed  approximately 
on  the  average  by  the  formula, 


(77)    e  =  e0 10  6517    in  meters,  e  =  e0  21381  in  feet. 

It  will  be  shown  that  the  barometric  pressure  diminishes  by 
a  similar  law, 


(78)  B  =  Bo  10     1840°  in  meters,  B  =  B0 10    60367  in  feet. 

p 

By  combining  these  in  the  ratio  5-  it  becomes, 

(79)  4-  =  -^  1Q~10091  in  meters,  ~-  =  -~  lO'33^  in  feet. 

£>  £>Q  &  JL>O 

These  can  be  reduced  from  the  common  base  10  to  the 
Naperian  base  e  by  the  modulus  M  =  0.43429. 

10091  X  M  =  4383,  33108  X  M  =  14378. 
The  expression  for  the  integral  mean  from^Zpjto  z  is 

(80)  j8  =  — - —  0.378  ~  /  V  ^ssa  j  z  for  the  metric  system. 

Z    —    ZQ  £>Q   ^ZQ 

I  en       Cz  -  —— 

(81)  j8  =  —      -  0.378  -£•  /    e  14378  d  z  for  the  English  system. 

Z    —    ZQ  £>Q  •'ZQ 

These  can  be  developed  in  a  series,  as  shown  in  the  "Report 
on  the  International  Cloud  Observations,"  UJ3.JSLJ3.,  1§98Z_ 
page  491,^" 


FORMULAS   FOR  MIXTURE    OF    GASES  23 


metnc- 


i  /  z  \z          ~] 

~T\  \4383 /  ^  *  ' '  J 
(83)  ft  =  0.378-1  [l  -  4l  yb*  )  +  4t  (74^)2- 

JJQ    —  ^i    ^J-TcO/O  /  ol    ^XTtOiO' 

1       /        0        V  IT 

-4!Vl4378)+-"JEnSllsh- 


That  is  to  say,  having  the  vapor  pressure j^  and  the  barometric 

pressure  BQ  at  the  surface  one  can  compute  the  average  value 

& 
of  the  integral  of  the  term  0.378  ^  up  to  the  height  z.    It  is 

commonly  impractical  to  measure  the  values  of  e  and  B  at 
several  points  in  the  atmosphere,  and  for  many  computations 
this  method  of  mean  integration  upward  from  the  surface  is 
quite  sufficientjpr_pjractical  work.  Also,  it  is  a  very  expeditious 
method  when  using  the  humidity  table  92,  page  548,  for  metric 
measures  and  Table  19  of  the  "Barometry  Report/'  U.  S.  W.  B., 

o 

1900,  page  108,  for  English  measures.     The  mean  values  of  -g 

for  the  air  column  is  often  taken  as  the  arithmetical  mean  of 
the  observed  values  at  an  upper  statiqn_z  and  a  lower  station  J;Q.____ 
In  the  case  of  balloon  and  kite  ascension,  the  registered  relative 
humidity,  temperature,  and  pressure  can  be  computed  to  the 
integral  mean  value  required.  When  only  temperature  and 
pressure  are  registered,  this  correction  to  the  density  of  the 
atmosphere  is  not  available  in  the  hypsometric  formula. 

The  General  Formulas  for  the  Mixture  of  Gases 

The  general  principles  controlling  the  mixture  of  gases  are 
so  often  useful  in  meteorology  that  it  will  be  convenient  to  collect 
together  the  common  formulas  expressing  the  several  processes. 
It  will  now  be  proper  to  pass  from  the  system  of  M.  K.  S.  units 
to  the  system  of  C.  G.  S.  units,  and  to  bring  forward  the  terms 


24  METEOROLOGICAL  CONSTANTS  AND  FORMULAS 

applicable  to  the  thermodynamics  of  the  atmosphere  and  the 
kinetic  theory  of  gases.     We  adopt  the  following  notation: 

a  =  atomic  weight. 

K 

m  =  molecular  weight  =  -5-. 

K.  f 

(84)  K  =  the  absolute  gas  constant  =  m  R  =  ~  —  . 

*• 


Compute  K  in  the  (C.  G.  S.)  system,  using  the  gas  hydrogen, 
mH  =  2,  pH  =  0.000089996.     Log  =  5.95422  -  10. 


82482000 

=  1-9708  small  calories  or  therms. 


Using  the  values  for  air  m0  =  28.736,  p0  =  0.00129305,  the 
same  result  is  obtained.  This  formula  applies  to  the  three 
systems  and  to  all  gases.  The  density  and  molecular  weight  of 
hydrogen  are  related  to  those  of  other  gases  so  that, 

(85)  m  pH  =  mH  p, 

and  for  this  reason  hydrogen  is  the  standard. 

n  =  the  number  of  molecules  in  a  unit  volume. 
N  =  the  number  of  molecules  in  a  V- volume. 

M  =  N  m  =  Mass. 

(86)  Number.      N  =  —  =  V  n  =  ^~ 

m  K  1 

P  V  m       P  V 

(87)  Mass.          M  =  N  m  =  V  nm  =        T    =  TT^  = 

P       M_        PM 

R^T  '  nm  ~  RT  p' 

M        M        MKT 

(88)  Volume.      V  =  M  v  =  -  -=-  = 

p        nm       m    P 


FORMULAS   FOR   MIXTURE   OF   GASES  25 

(89)  Pressure.    jP^  =  -  -=—  =  nK  T  = =-  =  ~RT  = 

nmRT  =  p  R  T. 

(90)  Density.         p  =  —  =  WW  =  TT=  —  yr  =  n~^. 

,mv  „••.  1          1          F       A"  T      RT 

(91)  Volume  of  unit  mass,     v  =  —  =  -   -=—  =  —  —  =  — — . 

p        nm       M       m  P       P 

(92)  Constant.      R  =  —  =  —  -  =  =— -  =  -^-. 

m        M  T       T  nm       T  p 

Referring  to  the  standard  gas  hydrogen  there  are  some  special 
values  for  the  gas  constant. 

Logs. 
n  p        ~vH       ~  m  gram  cm.2 


(94) 


(95)    K    =  ^^*  =  ^^  =  82482000  gram  X 
PH  PH  J-  o 


sec.2    temp. 


If  7?r  is  the  gas  constant  for  the  heat  energy  at  T, 
Rm  is  the  gas  constant  for  the  molecular  energy, 
Ra  is  the  gas  constant  for  the  atomic  energy.  Then, 

3          r     the  mean  kinetic  energy 
( Ub )  KI  =  KT  +  Km  -\-  Kn  =  —K  =  —  —  — =- — =— - 

2          T       absolute  temperature  . 

(97)  Rz  =  ^  Ri  —  R  =  Cv  the  specific  heat  at  constant  volume. 

o 

(98)  R3  =  -  RI  =  Cp  the  specific  heat  at  constant  pressure. 

o 

These  are  related  to  the  potential  and  kinetic  energies  in  the 
following  relations,  and  thence  to  the  specific  heats  at  constant 
pressure  and  constant  volume. 


26  METEOROLOGICAL  CONSTANTS  AND  FORMULAS 


Inner  Potential  Energies 

(99)  /  =  Jm  -f  Ja*    The  inner  potential  energy  of  molecules 

and  atoms. 

o  o      r>          o 

(100)  Jm=-Pv=-  —  =  -  RT.      Inner  molecular  potential 

w  &   p       & 

energy. 
/0  =  Inner  atomic  potential  energy. 

2   — 
/o  =  ^  7^  (inner  viriol).     Initial  inner  potential  energy. 

o 

(101)  V  =   -  I  S  (Xx  +  Yy  +  Z.)    the  mean   viriol  for   the 

force  (X.  Y.  Z.). 

Inner  Kinetic  Energies 

(102)  H  =  Hm  -f  Ha.    The  inner  kinetic  energy  of  molecules 

and  atoms. 

Hm=  Inner  molecular  kinetic  energy. 

Ha  =  Inner  atomic  kinetic  energy. 

Total  Heat  and  Work  Energies 

Q  =  The  heat  or  total  inner  energy  =  Cv  T. 
W  =  the  work  or  total  external  potential  energy. 

(103)  Q  =  J  +  H  +  W  =  (|  £1  -  R)   T  +  /0.     Total   inner 

energy. 
p  2  — 

(104)  W  =  Pv  =  --  =  RT  =  -Ve    (outer    viriol).      External 

p  o 

potential  energy. 

*Jm  and  To  relate  to  the  trifling  rearrangements  of  parts  which  are  the  only 
changes  that  can  occur  in  ordinary  chemical  and  physical  reactions.  We  can 
not  attack  the  enormous  stores  of  energy  shut  up  within  the  atoms. 


INNER   KINETIC   ENERGIES,   WORK,   AND   HEAT  27 

(105)  U  =  J  -  /o  +  W  =  Jm  +  Ja  -  /o  +  W.      Potential 

energy. 

(106)  U  —  W  =  J  —  JQ.     Accession  of  inner  potential  energy. 

(107)  V  =  H  =  -  l-  S  (Xx  +  Yy  +  ZJ  =  |  U  =  mean  viriol 

or  work  done. 

The  gas  constants  are  again  denned  thus: 

TJ 

(108)  RT=  7p  (heat)  =  the  ratio  of  the  inner  kinetic  energy  to  T. 

(109)  Rm=  j^-^mq2  where  q2  =  the  mean  square  velocity. 

(110)  Ra  =  -^  -n  m^i2,  n  =  the  number  of  atoms  in  a  molecule. 

1    2> 

The  Specific  Heats  of  Monatomic  Gases 

5  55 

(111)  Cp=  -Rl-R  +  R==R2  +  R  =  -Rl  =  -R. 

7?      2  7?        2  7? 
=  3*1  =  3  R* 

/  ~*  -« <-\\     /~*at  3?  7?    Z?  3?     .  —    J2     -        3?    1? 

3 

T>     T)  T> 

2  ^  _  Kl  _  y^2 

,    ON  Cp      RZ  +  R     5 

(113)  pr-  =  — ^ —  =  r-  =    1.67    =   k    for    monatomic    gases 

(mercury). 


c-c  k  - 

'  (20)' 


(116)  Cp  -  Cv  =  R  =      ^.     Compare  (15),  (16). 

O 


2        CD 


28  METEOROLOGICAL  CONSTANTS  AND  FORMULAS 

,11ftv    Ja  Jm  3  Cp-Cv       5Cv-3Cp 

(118)  -Q-  -Q-          -  2  — oT  ~2C~V       • 

The  above  formulas  are  in  mechanical  units,  and  they  can  be 
transposed  into  heat  units  on  multiplying  them  by  A  =  the  heat 
equivalent  of  work  which  is  A  =  0.00002343  in  ,the  C.  G.  S. 
system  by  Table  2. 

The  Fundamental  Laws  of  Physics 

(119)  Boyle's  (Mariotte)  Law.    P0vQ  =  Pv=  (-}    =  R0T  = 

\p/  T 

constant  for  constant  T. 

CT)  \  ~D 

•=rj   =  —  =  constant  for  constant  v. 

Pv 

(121)  Boyle-Gay  Lussac  Law.      -=r  =  R  =  constant   in   per- 

fect gases. 

(122)  Avogadro's    Law.     N  =  —  =  T^  =  constant  for    (P. 

m       A  i 

V.    T.)  constant. 

(123)  Clausius'    Law.      J  mi  <?i2  =  f  mz  qf  —  constant   kinetic 

energy. 

(124)  Dalton's   Law.      P  =  PI  +  P2  +  .  .  .  =  f  (J  mi  qS  + 

J  mz  qz*  +  ...).     Pressures. 

In  ordinary  gases  as  distinguished  from  perfect  and  ideal 
gases,  all  these  formulas  are  more  or  less  defective  on  account  of 
the  internal  action  of  the  atoms  and  the  molecules  upon  each 
other  under  the  stresses  of  electrical  and  other  mechanical  forces. 
Many  formulas,  with  constant  coefficients  and  exponents,  have 
been  devised  to  take  account  of  these  physical  variations,  but 
they  will  not  be  further  mentioned  in  this  place. 

The  Formulas  for  the  Mixture  of  Several  Gases 

(125)  Pressure.  P  =  PRT  =  ~-  =  nmR  T  =  n  K  T  = 

K  .        RTM 


FORMULAS  FOR  THE  MIXTURE  OF  SEVERAL  GASES      29 

(126)  P  =  P,  +  P2  +  P3  +  .  .  .   =   (»!  +  m  + 


(127)  Pi  =         .     *'    -  P. 


n  n 

(128)  Pi  :  P2  :  P3  .  .  .  =  Vl  :  V2  :  V3  .  .  .  = 

Rt-Mi:RtM9:KiM^ 

(129)  P  V  =  (MlRl  +  M2R2  +  .  .  .)  T  = 


(130)  Mass.  =     ,         ,          >       ^^ 

m        mi       nh        m$ 


...     M1+M2+M3+. 


n  Mi    M*    Ma 


(132)  n^,n,...,,...  =  P,,P,,P,... 

(133)  Density.          n  m  =  HI  mi  +  n2  nh  +  ns  m3  +  .  .  .  = 

PI  +  P2  +  PS  +  .  .  .   =  P  =  —  -. 


(134) 


11 

(135)  Gas  Constant.    R  - 


.  .  .   K 


n2m2 


*  RI  R2  R3)  etc.,  in  these  equations  refer  to  the  various  gas  coefficients  of 
the  several  gases. 


30  METEOROLOGICAL   CONSTANTS   AND   FORMULAS 

(136)  Inner  Energy.  M  U  =  M l  Ul  +  M 2  £/2  +  M 3  U3  +  . . .  . 

(137)  Entropy.         M  S  =  M^  Si  -f  M2  S2  +  Mz  S3  +  .  . .  . 

(138)  Concentration,     c  =  ~  — — '-LL-. 


(139)  c  R  = 


The  Kinetic  Theory  of  Gases  for  the  Atmosphere 

The  various  formulas  involving  specific  heat  can  all  be  de- 
duced from  the  kinetic  theory  of  gases,  and  it  is  therefore  desir- 
able to  have  at  least  approximate  values  of  the  constants  of  the 
principal  gases  which  are  the  constituents  of  the  atmosphere. 
These  are  arranged  in  Table  7,  so  that  the  formulas  from  which 
they  are  derived  suggest  by  definition  the  exact  meaning  of  the 
several  terms.  It  is  much  better  to  depend  upon  formulas  for 
defining  constants  than  upon  any  extended  verbal  description 
for  the  sake  of  accuracy  and  brevity.  On  the  other  hand,  it  is 
not  possible  to  study  any  advanced  research  problem  in  atmos- 
pheric physics  without  depending  upon  the  several  terms  in 
the  kinetic  theory  of  gases.  In  the  present  status  of  physics, 
research  is  attempting  to  make  out  the  connection  between  the 
theory  of  mechanical  collisions  in  the  molecules  of  a  gas  and  the 
corresponding  dynamic  electric  and  magnetic  forces,  but  this 
investigation  is  incomplete. 

It  would  be  very  desirable  that  some  international  commission 
should  adopt  a  series  of  consistent  constants  for  the  terms  of 
Table  7,  in  order  that  all  computations  may  be  made  on  the 
same  basis.  At  present  there  are  small  variations  in  the  values 
in  consequence  of  adopting  slightly  different  fundamental  con- 
stants from  which  the  others  are  derived.  It  is  probable  that 
sufficient  agreement  exists  among  chemists  and  physicists  as  to 
these  elementary  constants,  in  order  to  make  this  a  practical 
proposition. 


KINETIC   THEORY   OF   GASES 


31 


.a  o> 

118 


.; 

O  O 


•5.$ 

ir 

*H     "O 


a=a 


H  ':!TOiii4i  ii 

:  II  «     5^.y?-tSSSS-eoiS     tec 


I   I 


IS    C 
10     £ 


«    fl    B 
.2    S    3 

Hi 

n  v  P 


32 


METEOROLOGICAL   CONSTANTS   AND   FORMULAS 


This  table  was  computed  for  the  constants, 
K  =  82481110.     Absolute  gas  constant,  dynes/cm.2 
•*-    P  =  1013235.     One  atmosphere  in  dynes/cm.2 

-j-  =  41852800.     Mech.  equivalent  heat  in  ergs. 

It  is  very  desirable  that  the    International  Meteorological 
Committee  should  fix  standard  values  throughout  the  table. 

The   Temperature   and  the   Temperature  Gradients  Observed   at 
Different  Elevations  in  the  Free  Air 

The  actual  temperature  of  the  atmosphere  at  any  point  is 
the  resultant  of  the  force  of  gravitation  as  balanced  by  the 

TABLE  8 

EXAMPLES  OF  TEMPERATURE  AND  TEMPERATURE  GRADIENTS  AT 
DIFFERENT  ELEVATIONS 


Lindenburg 

Lindenburg 

Atlantic  Ocean 

Victoria  Nyanza 

Station 

Apr.  27,  1909 

May  5,  1909 

Sept.  25,  1907  Lat. 

Summer,  1908 

Lat.  +  52° 

Lat.  +  523 

+  35°Long.  +  36J 

Lat.  0J 

Height  z 

AT 

AT 

AT 

AT 

in  meters 

T 

1000 

1000 

1000 

1000 

18000 

226  5 

4-30 

190  5 

—  6  5 

17000 

223.5 

+    1.0 

222.9 

+    2.7 

197.1 

-5.5 

16000 

222.5 

4-18 

220.2 

+    0.1 

202.6 

-  4  2 

15000 

220.7 

4-10 

220.1 

+    0.2 

206  8 

—  4  0 

14000 

219.7 

+    28 

219.9 

-    0.2 

210.8 

-52 

/    13000 

202.7 

-    0.8 

216.9 

+    5.6 

220.1 

-    0.2 

216.0 

-6.6 

12000 

203.5 

-    6.4 

211.3 

-    1.7 

220.3 

-    7.8 

222.6 

-8.8 

11000 

209.9 

-    8.8 

213.0 

-    8.4 

228.1 

-    7.9 

231.4 

-7.5 

10000 

218.7 

-    9.9 

221.4 

-    9.1 

236.0 

-10.0 

238.9 

-7.2 

9000 

228.6 

-11.9 

230.5 

-    8.8 

246.0 

-    7.7 

246.1 

-4.6 

8000 

240.5 

-    8.8 

239.3 

-    8.4 

253.7 

-    7.4 

250.7 

-7.3 

7000 

249.3 

-    7.0 

247.7 

-    7.9 

261.1 

-    6.3 

258.0 

-5.4 

6000 

256.3 

-    6.6 

255.6 

-    7.3 

267.4 

-    6.6 

263.4 

-5.8 

5000 

262.9 

-    6.2 

262.9 

-    4.2 

274.0 

-    4.5 

269.2 

-6.5 

4000 

269.1 

-    6.6 

267.1 

-    5.8 

278.5 

-    3.4 

274.7 

-6.1 

3000 

,  275.7 

-    5.3 

272.9 

-    3.9 

281.9 

-    5.2 

280.8 

-7.6 

2000 

281.0 

-    5.5 

276.8 

-    2.9 

287.1 

-    3.4 

288.4 

-7.8 

1000 

286  5 

7  9 

279  7 

—    1  6 

290  5 

—    6  0 

296  2 

000 

294.4 

281.3 

296.5 

hydrostatic  pressure,  the  circulation,  and  the  radiation.     It  is 
the  most  important  element  to  be  observed,  and  from  it  all  the 


TEMPERATURE    GRADIENTS   IN   FREE   AIR 


33 


other  terms  can  be  computed,  provided  the  velocity  and  the 
vapor  pressure  are  also  given  by  the  observations.  In  order  to 
have  the  data  in  concrete  form  so  that  the  formulas  may  become 
practical,  four  examples  are  taken  from  the  observations,  at 
Lindenburg,  Germany,  in  the  Tropic  North  Atlantic  Ocean,  and 
at  Victoria  Nyanza,  Africa.  Table  8  records  the  height  in 
metersj^  the  absolute  temperature  T,  the  vertical  temperature 

^  T 

gradient  per  1000  meters  ,  and  Table  9  the  relative  humidity 

-LUUU 

R.  H.,  and  the  vapor  pressure  in  millimeters  e.  The  latter  is 
computed  by  taking  from  the  Tables  of  vapor  pressure  in  saturat- 
ed air  at  given  temperatures  the  dew-point  vapor  pressure,  and 
multiplying  by  the  relative  humidity.  The  Smithsonian  tables 
have  been  extended  to  include  approximately  the  vapor  pressure 

TABLE  9 

EXAMPLES  OF  THE  CORRESPONDING  RELATIVE  HUMIDITY  AND  VAPOR 
PRESSURE  AT  DIFFERENT  ELEVATIONS 


Height  z 

R.H. 

e 

R.H. 

e 

R.H. 

e 

R.H. 

e 

19000 

per  cent. 

mm. 

per  cent. 

mm. 

per  cent. 

mm. 

per  cent. 

mm* 

18000 

43 

0.020 

33 

0.000 

17000 

43 

.004 

47 

0.014 

33 

.000 

16000 

. 



43 

.012 

47 

.012 

33 

.001 

15000 

• 



44 

.011 

47 

.012 

34 

.002 

14000 

44 

.010 

47 

.012 

34 

.003 

13000 

61 

'6.002 

44 

.007 

47 

.012 

34 

.004 

12000 

61 

.002 

43 

.004 

47 

.012 

34 

.010 

11000 

61 

.043 

43 

.004 

47 

.027 

34 

.029 

10000 

61 

.012 

43 

.012 

47 

.063 

34 

.058 

9000 

61 

.039 

43 

.033 

47 

.183 

34 

.134 

8000 

62 

.135 

44 

.084 

48 

.402 

35 

.218 

7000 

64 

.284 

45 

.209 

51 

.843 

36 

.449 

6000 

69 

.741 

47 

.473 

54 

1.544 

41 

.831 

5000 

72 

1.400 

50 

.972 

56 

2.750 

50 

1.667 

4000 

67 

2.370 

51 

1.420 

58 

3.910 

73 

3.770 

3000 

67 

3.710 

42 

1.910 

66 

5.600 

69 

5.440 

2000 

70 

5.590 

41 

2.460 

65 

7.770 

60 

7.800 

1000 

63 

7.250 

60 

4.390 

72 

10.700 

57 

12.030 

000 

54 

15.620 

58 

4.730 

78 

16.770 

at  very  low  temperatures.     These  values  of  T  and  e  will  be  used 
in  illustrating  the  barometric  reduction  formulas.     By  formula 


34  METEOROLOGICAL  CONSTANTS   AND   FORMULAS 

(22)  the  adiabatic  temperature  gradient  per  1000  meters  is 
-9.869°  C.,  or  per  1000  feet  -5.415°  F.  In  Table  8  it  is  seen 
how  widely  the  actual  temperature  gradients  differ  from  this  value, 
and  it  is  this  circumstance  that  compels  us  to  reconstruct  the 
entire  range  of  standard  thermodynamic  formulas,  in  order  to 
adapt  them  to  practical  work  in  the  earth's  atmosphere.  The 
slow  progress  of  meteorological  physics  is  due  to  this  difference 
of  gradient  more  than  to  any  other  cause.  The  temperature 
gradients  are  incessantly  varying  in  the  atmosphere  from  large 
temperature  falls,  —  A  jT,  to  considerable  temperature  gains, 
-f  A  T.  This  change  of  sign  gives  rise  to  the  subject  of  the 
inversion  of  temperature  of  which  examples  occur  often  at  night 
near  the  ground,  and  usually  in  the  isothermal  region. 

The  values  of  the  vapor  pressure  below  T  =  273°  are  those 
of  the  Smithsonian  Tables  extended. 

The  Temperature  Gradient  in  a  Plateau  from  the  Sea  Level  to  the 
Surface  of  the  Ground 

An  important  part  of  barometry  is  the  determination  of  the 
temperature  gradient  within  the  land  mass  forming  a  plateau 
region,  as  in  the  Rocky  Mountain  district  of  the  United  States, 
by  means  of  which  the  pressure  observed  at  a  station  on  the 
surface  may  be  reduced  to  the  sea  level,  in  order  to  be  combined 
with  those  stations  having  low  elevations,  so  as  to  make  a 
synchronous  map  of  storm  conditions  for  the  entire  country. 
This  problem  is  one  of  considerable  difficulty,  and  it  must  be 
solved  in  accordance  with  the  prevailing  local  conditions,  so 
that  no  fixed  rules  can  be  given  for  its  treatment.  A  very 
extensive  reduction  for  the  United  States  is  found  in  the  "Report 
on  Barometry,"  already  mentioned,  but  its  leading  principles 
can  be  briefly  summarized.  Having  low-level  stations  in  the 
eastern,  central,  and  Pacific  districts,  the  problem  is  to  connect 
up  the  stations  on  the  plateau  at  different  elevations  above  the 
sea  level,  by  means  of  the  average  temperature  gradient  within 
the  land  mass,  which  is  very  different  from  the  gradient  above 
the  plateau  in  the  free  air.  The  first  step  is  to  construct  from 
the  available  data  approximate  temperature  gradients,  which 


TEMPERATURE  GRADIENT  IN  A  PLATEAU          35 

can  be  used  for  short  distance  reductions,  in  longitude, in  latitude, 
and  on  the  vertical.  Then  certain  reference  vertical  lines  are 
chosen,  as  the  intersection  of  the  parallels  and  meridians  for  each 
5-degree  interval,  and  for  the  planes  1000  feet  apart.  To  these 
points  are  reduced  the  stations  by  numerous  combinations,  so  that 
the  same  station  is  reduced  to  several  selected  reference  points. 
These  points  now  lie  on  vertical  lines,  and  each  line  of  tempera- 
tures may  by  plotting  be  extended  downward  to  sea  level.  The 
second  step  is  to  draw  the  sea-level  isotherms  between  the 
central  and  the  Pacific  districts,  joining  across  the  plateau 
region  by  the  most  probable  curves.  The  third  is  to  compare 
by  interpolation  these  horizontal  temperatures  with  those 
obtained  from  vertical  extension,  and  by  mutual  adjustments 
the  two  sets  may  be  made  to  agree  harmoniously.  This  inter- 
locking of  a  horizontal  system  with  a  vertical  system  is  able  to 
produce  by  mutual  checking  a  very  exact  agreement  between 
the  two  sets.  In  this  way  the  sea-level  temperature  was  found 
beneath  the  plateau,  and  thence  the  temperature  gradients  in 
a  vertical  direction  were  computed.  These  gradients  differ 
greatly  from  one  another  in  different  parts  of  the  plateau,  they 
differ  from  month  to  month  at  the  same  place,  and  there  is  no 
fixed  gradient  which  can  be  used  at  any  given  station.  The 
station  gradients  fall  into  distinct  classes  in  the  several  parts  of 
the  plateau  in  respect  to  the  yearly  variations,  and  diagrams 
were  constructed  to  serve  for  the  individual  stations.  The 
monthly  gradient  may  be  roughly  summarized  for  comparison 
with  the  free-air  gradients. 

A  T  °C.          AT  °F. 

The  reduction  from  7^7-          -  to  -    _   ,  --  is  effected    by 
100  meters        100  feet 

the  factor—1—  =  0.55.     This  is  from—1—  X  ^~  =  p^0'55' 

This  plateau  gradient  is  only  37  per  cent,  of  the  free  air  adiabatic 
gradient.  Extensive  reduction  barometric  tables  were  con- 
structed for  many  stations  in  the  United  States,  which  are  used 
in  compiling  the  weather  forecast  charts,  of  which  further  mention 
will  be  made. 


36 


METEOROLOGICAL  CONSTANTS  AND  FORMULAS 


TABLE  10 

THE  MEAN  MONTHLY  TEMPERATURE  GRADIENTS  IN  THE  ROCKY  MOUNTAIN 
PLATEAU  OF  THE  UNITED  STATES 


A  27100 

Jan. 

Feb. 

March 

Apr. 

May 

June 

Per 
100  ft. 
Per 
100m. 

-0.191 
-0.348 

-0.174 
-0.317 

-0.145 
-0.264 

-0.202 
-0.368 

-0.194 
-0.353 

-0.215 
-0.391 

A  77100 

July 

Aug. 

Sept. 

Oct. 

Nov. 

Dec. 

Year 

Per 
100  ft. 
Per 
100m. 

-0.175 
-0.319 

-0.179 
-0.326 

°F 
-0.168 

-0.306 

-0.191 
-0.348 

-0.181 
-0.329 

-0.187 
-0.340 

op 
-0.202 

-0.368 

The  Integral  Mean  Temperature  and  Gradient 

In  reducing  pressures  from  one  level  to  another  it  is  necessary 
to  know  the  mean  temperature  of  the  actual  air  column  in  the 
free  air,  or  in  the  hypothetical  air  column  within  a  plateau,  or 
between  a  mountain  summit  and  the  sea  level,  or  other  plane  of 
reference.  These  are  found  from  the  summation  of  the  tem- 
peratures at  several  levels  as  in  Table  8,  and  dividing  the  sum  by 
the  number  of  the  strata  taken. 


(140) 


from  (23). 


For  two  strata  it  is  the  arithmetical  mean, 
(141)     7\0  =  J  (71!  +  To). 

For  example,  in  the  balloon  ascension,  April  27,  1909,  sum  the 
Tz  through  the  several  strata;  also,  only  the  top  and  bottom  of 
the  same  thick  layer,  as  indicated. 


INTEGRAL  MEAN   TEMPERATURE   AND   GRADIENT  37 


Layer  in  meters 

-s  r 

z 

i  (TJ.  +  r0) 

000  to    4000 

281.34 

281.75 

000  to    9000 

264.43 

260.15 

000  to  13000 

248.51 

243.55 

It  is  not  usually  sufficient  to  take  the  mean  value  of  the  top 
and  bottom  temperature  of  a  thick  layer  for  the  mean  tempera- 
ture of  the  column,  and  the  error  of  reduction  is  dependent  upon 
the  discrepancy  between  Tm  and  TiQ.  If  the  gradient  is  uniform 
between  two  strata  Tm  =  Tw,  and  the  difference  vanishes.  The 
length  of  vertical  distance  that  permits  7\0  to  be  used  depends 
upon  local  temperature  distributions,  and  each  case  must  be 
carefully  examined.  The  same  rule  applies  to  the  vapor  pressure 
e,  and  any  other  meteorological  element.  The  determination  of 
the  integral  mean  with  accuracy  is  one  of  the  hardships  of  practi- 
cal meteorology,  upon  which  a  large  amount  of  labor  is  necessarily 
expended. 

The  Virtual  Temperature  Tr. 

It  is  sometimes  convenient  to  combine  the  actual  mean 
temperature  Tm  with  the  expression  for  the  vapor  pressure 

p 

term  0.378-^  to  form  the  so-called  virtual  temperature  Tr,  by 
the  formula, 

(142)  2T-  =  Tw(l  + 0.378  J) 

The  barometric  reduction  can  then  be  carried  forward  as  if 
the  dry  air  and  the  aqueous  vapor  were  compounded  in  one  gas 
whose  equivalent  temperature  is  Tr. 

/I   rrt 
^jT. 

The  ratio  of  the  change  of  temperature  d  T  to  the  prevailing 
temperature  T  is  related  to  the  logarithm  in  the  follov/ing  useful 
auxiliary  formulas,  which  are  often  needed  in  substitutions. 


38  METEOROLOGICAL   CONSTANTS   AND   FORMULAS 

They  are  applicable  through  such  strata,  thin  or  thick,  as  have 
uniform  temperature  gradients,  whether  the  temperature  in- 
creases or  diminishes  in  a  vertical  direction.  They  are  given 
for  the  adiabatic  and  the  non-adiabatic  temperature  variations. 

/•<    ^o\  I®         a  a  0  1  a  /I  T1  1  T>    \ 

(143)  /  -=-  =  —T =  M  loS  r"  =  M  (loS  r«  ~  loS  ro). 

J      1  a  J-  aO  M-  1  o         M 

(144)  f^  =  r'~r°  =  i  log  p  =  ^  (log  Tx  -  log  To). 
J      1  lio  M          1  Q       M 

Ti-T0  1 


10 


1 

(n-  1)  M    gR0  ~  (n-  1)  M 


The  Temperature  Variations  and  the  Specific  Heat 

It  is  convenient  to  make  the  transfer  from  the  non-adiabatic 
temperature  loss  to  the  adiabatic  temperature  loss,  in  connection 
with  the  specific  heat,  by  using,  as  in  (13), 

Ta-T0 


(146)  »i  Cpa  (T,  -  To)  =  Cpa  (Ta  -To)  =  -  go(zi-z0). 

(147)  m  Cp10  (T,  -  To)  =  Cp10  (Ta  -  To)  =  Pl  ~  P° 

Pi0 

These  formulas  will  be  fully  illustrated  in  a  later  chapter. 
It  is  evident  that  many  combinations  can  be  made  by  employing 
the  formulas  (143)  to  (147),  and  they  are  very  practical  in  de- 
veloping the  formulas. 

T          f  j-          r^°         Bo  I  0£r  Zi  —  Z0\ 

Transformation  of  -5-  =  -^  f  1  +  1.25  — ^ — J. 

The  introduction  of  the  plateau  effect  upon  gravity  in  (69) 
has  its  parallel  in  the  effect  upon  the  barometric  pressure,  which 
is  similarly  modified.  We  have  for  both  cases, 


(i«>  «.  -i.  (i  + 


GENERAL  BAROMETRIC  FORMULA  39 

Since  1  +  1.25  —  ~^~  is  a  small  variation  from  unity,  the 
K 

general  formula  is  applicable, 

(150)  Com.  log  (1  +  x)  =  M  (x  -  %  x*  +  %  x*  -  .  .  .). 

Passing  to  common  logarithms  (149)  becomes  on  neglecting 
the  powers  above  x, 

(151)  log  J  =  log  |°  +  log  (l  +  1.25  *-^°)  , 

.     Bo   ,   M  1.25  , 

=  log  -g   +  --  £—   (2i  -  Z0). 

It  will  be  found  in  the  barometric  formula  that, 

P  '  B 

(152)  Zi  —  z0  =  K  log  -5-  (approx.)  =  K  log  -_-  (approx.).     Hence, 

/ieo\        1         ^°          1         -^°      i      1    nt-M  K  BQ 

(153)  log     -  =  log  -     +  1.25  --  log  -   , 


logj°  (1  +  0.00157)  =  log  ~°  (1  +  y). 


The  General  Barometric  Formula 

The  several  auxiliary  formulas  now  deduced  make  it  very 
simple  to  derive  the  barometric  reduction  formula  connecting 
together  the  heights  (z\  .  z0)  and  the  mercurial  pressures  (5i,  B0). 
From  (41)  the  differential  pressure  is, 

(154)  -  dP  =  p.gdz.          Divide  by  P, 

(155)  -d-j-=^gdz.  Divide  by  |, 

d  P        P 

(156)  -  -5-    .   -  =  g  .  d  z.     Substitute  from  (75), 

*          P 


(157)     -  --   .   -°    •    l  +  0.378         =  g»  d  z. 

po  1  o 


40  METEOROLOGICAL  CONSTANTS  AND  FORMULAS 

Substitute  P0  from  (49)  and  g+f  from  (63),  (64), 
_  dP       Bo  Pm  go  —  d    i   Q  370  _£_\  = 

X  Po  J-  0    \  -t>Q/ 

go  (1  -  0.0026 cos2(j>)  (l  -  -jj{}  d z. 
Pass  to  common  logarithms  by  the  factor  -^.  and  integrate, 


/IKON    i      -^°       -^Q  Pm       Tm  /  e\ 

(159)    log  -p-   .   -vp-     .   —  (  1  +  0.378  -=-  )     = 

°  r  M   po  1  o     \  JDQ/  m 


e 
=- 

o  DQ     m 

-  0.0026  cos 


The  last  gravity  form  is  from  (67).     The  constant 

(160)  K  =  2~  =  18400  (Metric),  and  JRTi  =  60367.7  (English). 
M  po 

Tm  is  the  mean  temperature  of  the  column  (140),  and  the 
integral  mean  of  (  1  +  0.378-^-)  is  accomplished  by  the  observa- 

\  JJQ/ 


tions  along  the  column,  or  by  integrating  from  the  surface  by 
(82)  and  (83)  .     Substituting  (153) 

(161)    log  ~  (1  +  0.00157)  K(l+  0.00367  9)  (l  +  0.378  £} 

&  \  -L>0/  m 

Barometer.  "         Const.         Temperature.  pSsure. 


(1  +  0.0026  cos  20)     l  +    l        °     =  zi  -  Z0. 


Gravity  in  Gravity  in 

Latitude.  Height. 


Substituting  the  numerical  values  and  combining, 
(162)    log  So  =  log  B  +  18429+67Y0°2C.+  o.003  < 


-  0.378^-)  (1  -  0.0026  cos  20).     Metric. 
(163)    log£0  =  logB  ~ 


-  0.378^-)  (1  -  0.0026  cos  20).     English. 


CORRECTIONS   TO   THE   BAROMETER  41 

These  can  be  expressed  in  the  general  form, 

(164)  log  B,  =  log  B  +  m  (1  -  0)  (1  ~r) 

=  log  B  +  m  —  m  ft  —  m  ?. 

In  view  of  the  uncertainty  attaching  to  our  knowledge  at 
any  time  of  the  distribution  of  the  vapor  pressure  in  the  air 
column,  it  is  desirable  to  keep  the  term  m  ft  separate  as  a  correc- 
tion to  the  difference  between  the  logarithms.  Similarly  the 
gravity  term  is  retained  by  itself  because  in  many  computations 
it  is  small  and  can  be  neglected.  Complete  reduction  tables  are 
given  in  the  "  Report  on  the  International  Cloud  Observa- 
tions" for  the  metric  system,  Tables  91,  92,  93,  and  in  the 
"  Report  on  the  Barometry  of  the  United  States,  Canada,  and 
the  West  Indies"  for  the  English  system,  Tables  13  to  21. 
From  these  logarithm  tables  many  forms  of  numerical  tables 
without  logarithms  can  be  constructed  for  special  purposes. 

Corrections  to  the  Barometer 

The  mercurial  barometer  requires  several  corrections  before 
the  pressure  can  be  used  in  practice. 

1.  Correction  to  the  Standard  Temperature.  The  instrument 
is  constructed  of  parts  whose  coefficients  of  expansion  with 
changes  of  temperature  are  not  the  same,  as  for  the  mercury 
and  the  brass  scale.  Adopting  the  notation, 

t  =  the  temperature  of  the  attached  thermometer. 
tm  =  the  standard  temperature  of  mercury,  0°  C.,  32°  F. 
ts  =  the  standard  temperature  of  the  brass  scale,  0°  C.,  62°  F. 
m  =  the  coefficient  of  expansion  of  mercury,  0.0001818  per 
degree  Centigrade,  0.0001010  per  degree  Fahrenheit. 

n  =  the  coefficient  of  expansion  of  brass,  0.0000184  per  degree 
Centigrade,  0.0000102  per  degree  Fahrenheit. 
The  accepted  formulas  are  as  follows  : 

(165)  Bn  -  B  =  -  B  (m~n\t  for  Bn  and  B  in  millimeters. 

1  +  m  t 


inches. 


42  METEOROLOGICAL   CONSTANTS   AND   FORMULAS 

The  English  form  reduces  to, 

(167)  Bn  -  B        -B  10978  +  1.112 /' 

The  necessary  reduction  tables  are  found  in  nearly  all 
compilations  of  Meteorological  Tables. 

2.  Correction  to  the  Standard  Gravity,  g&.     This  is  the  gravity 
variation  in  latitude  as  given  in  (63),  from  which  is  obtained, 

(168)  £45  -  B+  =  -  B+  (l  -  ~ }  =  -  B+  0.00260  cos  2<£. 

V          £45  / 

The  temperature  and  gravity  corrections  are  applied  as 
instrumental  corrections  to  the  actual  barometric  reading  at  a 
given  hour. 

3.  Correction  to   a  Standard  Barometer  or  Patron.    Each 
barometer  as  an  instrument  has  certain  minor  deficiencies  which 
cannot  be  readily  analyzed,  and  in  order  to  make  a  number  of 
barometers   homogeneous,   so   as   to   give   strictly   comparable 
pressures,  it  is  necessary  that  they  be  severally  standardized 
by  comparison  with  an  adopted  normal  or  patron  barometer. 
The  Kew  barometer  is  used  for  many  standard  comparisons, 
and  each  weather  service  keeps  its  own  standard  which  has  been 
carefully  compared  with  the  Kew  instrument.    Within  each 
national  service  the  barometers  are  compared,  and  an  instru- 
mental correction  is  given  for  each  barometer  before  sending  to 
a  station.     Sometimes  these  corrections  hold  steadily  for  long 
intervals,  and  sometimes  they  change  suddenly  and  erratically. 
Whenever  there  are  local  removals,  or  whenever  a  barometer  is 
cleaned,  its  correction  must  again  be  determined.     Frequent 
inspections  and  comparisons  with  a  portable  secondary  standard 
are  necessary  if  a  homogeneous  series  of  pressures  is  to  be  secured . 
It  is  not  possible  to  be  too  painstaking  in  respect  of  the  inter- 
barometric  corrections. 

4.  The  Station  or  Removal  Correction.     If  it  happens  that  at  a 
given  station  there  are  any  removals  of  the  barometer  from  one 
office  to  another,  as  so  frequently  happens  in  large  cities,  and  the 
elevation  is  thereby  changed  from  time  to  time,  it  is  necessary 
to  adopt  a  standard  elevation  for  the  station  and  reduce  the 


CORRECTIONS   TO  THE   BAROMETER  43 

series  of  readings  taken  at  any  other  height  to  this  level,  which 
will  persist  as  the  adopted  station  elevation  from  the  beginning 
to  the  end  of  the  service.  When  the  change  in  height  is  con- 
siderable these  corrections  depend  upon  the  temperature  in  the 
course  of  the  year.  A  correction  card  for  instrumental  and 
station  removal  errors  should  accompany  each  barometer,  and 
be  carefully  recorded  as  part  of  the  history  of  the  instrument. 
In  preparing  homogeneous  tables  of  pressure  for  use  in  solar 
physics  and  other  cosmical  problems,  it  is  indispensable  that 
all  barometer  readings  should  be  carefully  treated  in  this  manner. 
The  homogeneous  system  for  the  United  States  has  thus  been 
prepared  by  the  author  to  cover  the  barometric  pressure,  the 
temperature,  the  vapor  pressure,  and  the  precipitation  from  the 
year  1871  to  date,  and  the  published  data  of  the  Weather  Bureau 
are  all  on  that  basis.  Similar  homogeneous  data  are  being  pre- 
pared for  Argentina  and  for  other  countries. 

5.  The  Correction  from  the  Surface  Temperature  (t)  to  the  Mean 
Temperature  of  the  Air  Column  (8)  in  Barometric  Reductions.  It 
is  obviously  so  difficult  to  determine  the  relation  of  the  surface 
temperature  /  to  the  mean  air  column  temperature  within  a 
land  mass  0,  as  from  a  station  on  a  plateau  to  the  sea  level,  that 
a  special  study  was  made  of  this  subject  in  order  to  facilitate  a 
prompt  reduction  of  the  observed  pressure  to  the  corresponding 
sea-level  pressure.  These  are  needed  for  transmission  by 
telegraph  to  a  central  office  where  the  daily  weather  forecast 
charts  are  constructed.  Unfortunately  there  is  no  simple  rule 
connecting  /  and  0,  and  in  many  cases  the  difference  6  —  t  is 
very  variable.  Reduction  tables  are  first  computed  with  the 
adopted  elevation  H,  and  a  series  of  assumed  values  of  6  for 
several  barometric  pressures  in  steps  of  0.10  inch.  Then  the 
relation  between  /  and  0  having  been  found,  the  surface  tem- 
perature is  used  as  the  argument  for  the  table  in  place  of  6.  The 
practical  value  of  /  taken  in  the  United  States,  where  the  observa- 
tions are  made  at  8  A.M.  and  8  P.M.  daily,  is  the  mean  of  the  cur- 
rent dry-bulb  temperature  and  that  taken  twelve  hours  before. 
This  gives  a  fair  temperature  average  for  the  day,  and  it  tends 
to  eliminate  some  of  the  local  effects  of  passing  storms.  It  has 


44 


METEOROLOGICAL  CONSTANTS   AND   FORMULAS 


been  found  to  work  well  in  the  practice  of  ten  years.  In  order 
to  illustrate  the  differences  between  /  and  6  in  the  course  of  the 
year,  as  the  temperatures  change  from  summer  to  winter,  a 
few  examples  are  extracted  from  Table  53  of  the  Barometry 
Report,  where  the  heights  are  in  feet,  and  temperatures  are 
Fahrenheit. 

TABLE  11 
RELATION  BETWEEN  THE  SURFACE  TEMPERATURE  /  AND  THE  MEAN  0 


Boise, 

Salt  Lake 

Independence, 

Helena, 

Pike's  Peak, 

Battleford, 

Idaho 

City,  Utah 

Cal. 

Mont. 

Colo. 

Canada 

2739 

4366 

3910 

4110 

14111 

1608 

1 

e 

t 

e 

t 

9 

/ 

e 

t 

e 

t 

e 

-42 

-40 

-48 

-40 

-61 

-40 

-32 

-30 

-37 

-30 

-49 

-30 

-40 

-30 

-23 

-20 

-24 

-20 

-26 

-20 

-35 

-20 

-29 

-20 

-13 

-10 

-14 

-10 

-15 

-10 

-23 

-10 

-17 

-10 

-  3 

-  0 

-  5 

0 

-  3 

'6 

-  4 

0 

-12 

0 

-  6 

0 

7 

10 

5 

10 

6 

10 

7 

10 

-  2 

10 

5 

10 

17 

20 

14 

20 

15 

20 

20 

20 

5 

20 

16 

20 

27 

30 

24 

30 

24 

30 

29 

30 

10 

30 

27 

30 

37 

40 

34 

40 

35 

40 

38 

40 

16 

40 

38 

40 

47 

50 

44 

50 

48 

50 

46 

50 

22 

50 

49 

50 

57 

60 

55 

60 

60 

60 

55 

60 

30 

60 

59 

60 

67 

70 

66 

70 

71 

70 

66 

70 

38 

70 

69 

70 

77 

80 

76 

80 

81 

80 

76 

80 

47 

80 

79 

80 

87 

90 

86 

90 

92 

90 

86 

90 

56 

90 

89 

90 

97 

100 

96 

100 

102 

100 

96 

100 

99 

100 

Similarly,  the  relations  between  the  surface  /  and  the  mean 
free-air  temperature,  or  the  mean  plateau  temperature,  6,  have 
been  prepared  for  reductions  to  the  sea-level  plane,  the  3,500- 
foot  level,  and  the  10,000-foot  planes  for  over  200  stations,  so 
that  synchronous  charts  can  be  constructed  on  each  of  these 
three  planes  simultaneously.  Such  charts  were  prepared  for 
one  year,  in  part  by  telegram  and  in  part  by  card  reports,  so 
that  the  pressure  charts  could  be  studied  on  the  sea  level,  on 
the  mean-plateau  level,  and  in  the  two-mile  level.  These  com- 
parisons were  so  suggestive  and  instructive  in  respect  of  the 
progress  of  storms  and  the  areas  of  precipitation  as  to  make 
them  of  great  value  in  practical  forecasts  of  weather  conditions. 
The  trend  of  the  upper-level  isobars  shows  clearly  the  course  of 


CORRECTIONS    TO    THE    BAROMETER  45 

the  storm  track  for  24  to  36  hours,  whereas  the  sea-level  isobars 
have  very  little  evidence  of  this  kind.  This  is  because  the  closed 
isobars  on  the  sea  level  have  generally  opened  up  into  sweeping 
curves  on  the  two-mile  level.  Similarly,  the  rain  areas  are 
indicated  by  the  region  of  most  oblique  crossing  of  the  lower 
with  the  upper  isobars.  There  is  a  great  future  for  meteorology 
in  the  use  of  these  upper  level  charts. 

6.  The  Plateau  Correction  C  A0  H.     An  extensive  discussion 
of  the  reduced  pressures  on  the  sea-level  plane  showed  a  series  of 
plateau  differences  depending  upon  a  station  constant  C  =  0.001 
usually,  A  0  =  the  departure  of  the  monthly  from  the  annual  6, 
and  H  =  the  height  of  the  station  in  feet  in  units  of  1,000  feet, 
so  that  A  B  =  C.  A  6.  H.     This  plateau  correction  was  computed 
and  applied  to  all  the  plateau  stations  of  the  United  States.     It 
seems  to  take  account  of  the  effect  of  the  land  mass  in  the  course 
of  the  year  upon  the  temperature  distribution,  which  is  very 
complex  in  its  action. 

7.  The  Local  Correction  A  A.    After  the  corrections  above 
mentioned  have  been  applied,  there  are  still  a  few  stations  which 
require  a  small  correction  A  A  to  make  them  harmonize  with  the 
pressure  system  surrounding  them.     The  cause  is  still  obscure 
and  is  very  local,  possibly  due  to  the  wind  action  near  the  office. 

8.  The   Local    Vapor   Pressure   Correction.    The   prevailing 
relative    humidity    and    the  corresponding  vapor  pressure  are 
approximate  functions  of  the  temperature  in  each  locality,  so 
that  an  approximate  value  of  the  correction  to  the  barometer 
due  to  the  presence  of  the  aqueous  vapor  can  be  found  for  each 
station  and  applied  along  with  the  other  corrections. 

9.  The  Station  Pressure  Reduction  Charts.    It  should  be  noted 
that  all  the  barometric  corrections  have  been  made  in  terms  of 
the  surface  temperature  so  that  this  t  and  the  barometer  reading 
B.  when  corrected  for  the  several  instrumental  errors,  become  the 
arguments  for  the  reduction  to  any  plane.     In  practical  work, 
instead  of  corrections  from  the  station  to  the  sea  level  or  other 
plane  of  reference  being  furnished  to  the  several  stations,  there 
have  been  prepared,  for  the  arguments  (/,  B),  the  reduced  sea- 
level  pressure  at  once  in  a  sufficiently  expanded  form  of  tables  to 


46 


METEOROLOGICAL  CONSTANTS   AND   FORMULAS 


permit  of  quick  and  accurate  interpolation;  similar  tables  were 
provided  for  the  3,500-foot  and  the  10,000-foot  planes.  With 
these  auxiliary  station  tables  the  reduced  pressures  are  promptly 
obtained,  and  transformed  into  the  telegraph  cipher  code  for 
transmission  to  other  offices.  In  this  way  a  large  number  of 
stations  in  the  United  States  receive  in  several  telegraph  circuits, 
by  interchange  of  messages,  the  necessary  data  for  all  map  con- 
struction. Making  the  observations  at  8  A.M.,  75th  meridian 
time  in  all  districts,  for  example,  the  data  are  received,  recorded, 
interpreted  as  forecasts,  and  usually  retransmitted  to  all  parts 
of  the  country  within  two  hours,  or  by  10  o'clock. 

Examples  of  the  Barometric  Reduction  Tables 

"Barometry  Report,"  U.  S.  W.  B.,  1900-01,  Tables  13-21. 
" International  Cloud  Report/'  U.  S.  W.  B.,  1898-99,  Tables 
91-93. 

In  order  to  illustrate  the  barometric  formulas  in  practice,  the 
example  of  Santa  Fe,  N.  M.,  is  here  given. 

TABLE  12 

REDUCTION  TO  THE  SEA  LEVEL  BY  THE  W-TABLES  IN  LOGARITHMS 
SANTA  FE,   NEW   MEXICO 

Height  =  7013  feet.     Longitude,  105°  57'     Latitude,  35°  41' 


Arguments 

Jan. 

April 

July 

Oct. 

Year 

Station  B 
Temperature  .  .  .6 
Vapor  Pressure  CQ 

23.180  in. 
32.0°F. 
0.145  in. 

23.177  in. 
54.  3°  F. 
0.199  in. 

23.362  in. 
79.0°F. 
0.574  in. 

23.294  in. 
58.0°F. 
0.294  in. 

23.  248  in. 
56.0°F. 
0.274  in. 

Logarithm  log  B. 
Table  17  m 
Table  19C,  -  $m 
Table  20  —  ym 

Sum.  log  BO 

1.36511 
+  .11594 
-.00016 
-.00011 

1  48078 

1.36506 
+  .11090 
-.00022 
-.00010 

1.47564 

1.36851 
+  .10581 
-.00061 
-.00010 

1  .  47361 

1.36725 
+  .11011 
-.00033 
-.00010 

1  .  47693 

1.36638 
+  .11054 
-.00031 
-.00010 

1  47651 

Sea  level  ....  B0 

30.254 

29.898 

29.759 

29.987 

29.958 

Table  19A  Arg  I  . 
Tablel9BArg.II 

.0018 
.0014 

.0025 
.0020 

.0073 
.0058 

.0037 
.0030 

.0035 
.0028 

Reduction  B0—B 

7.074 

6.721 

6.397 

6.693 

6.710 

EXAMPLES   OF   BAROMETRIC   REDUCTION 
SECOND  FORM  OF  TABLE,  NUMERICAL 


47 


Table  21,  Sec.  I.  . 

7.028 

6.762 

6.487 

6.719 

6.742 

Table  21,  Sec.  II. 

-f.  064 

-.018 

-.041 

+  .004 

-.003 

—  /?w  —  Ym 

-.018 

--.022  . 

-.049 

-.030 

-.028 

Reduction  

7.074 

6.722 

6.397 

6  693 

6.711 

The  station  pressure  B  is  already  corrected  for  the  tempera- 
ture, gravity,  instrumental  and  removal  errors. 

For  the  argument  ^  (tSp  +  O  or  y2  (t8a  +  t8p),  take  0. 

For  the  arguments  (B.  0),  in  Table  17,  take  m. 

For  the  arguments  (B0.  e0)  in  Table  19  A,  take  Arg.  I  (below). 

For  the  arguments  (Arg.  I,  H),  in  Table  19  J5,  take  Arg.  II 
(below) . 

For  the  arguments  (Arg.  II,  m),  Table  19  C,  take  -  0  m. 
For  the  arguments  ($,  m),  in  Table  20,  take  —  f  m. 


Numerical  Form 

For  the  arguments  (H,  B  =  30  inches),  in  Table  21, 1,  take 
first  reduction. 

For  the  arguments  H,  B^  (approx.),  in  Table  21,  II,  proceed 
by  trials. 

Interpolate  the  correction  for  humidity  from  Table  21,  III. 

Interpolate  the  correction  for  gravity  from  Table  21,  IV. 

These  two  methods  work  very  rapidly  after  a  little  practice, 
and  the  reductions  are  valid  to  the  0.001  inch  of  pressure.  In 
order  to  illustrate  the  method  of  reduction  for  the  plateau  the 
following  example  is  given  in  Table  13. 

The  assumed  station  pressure  B  has  the  four  station  correc- 
tions applied.  The  body  of  the  reduction  table  was  computed 
for  assumed  values  of  6,  which  correspond  with  certain  surface 
temperatures  /,  computed  from  the  .consecutive  8  o'clock  pairs 
as  observed.  This  t  becomes  then  the  practical  argument  for 
station  reductions.  The  mean  annual  6  for  Santa  Fe  was  taken 
63°  F.,  and  A  B  at  any  time  is  the  variation  (0  —  63°).  The 


48 


METEOROLOGICAL   CONSTANTS   AND   FORMULAS 


TABLE  13 

SANTA  FE,  NEW  MEXICO 

REDUCTION  OF  PRESSURE  TO  THE  SEA  LEVEL,  THE  3,500-  AND 
10,000-FooT  PLANES 

I. — Reduction  to  sea  level 
Elevation,  7,013  feet.     Longitude,  105°  57'.     Latitude,  35°  41' 


Temp. 

Correction  for 

22.40 

22.60 

22.80 

23.00 

23.20 

23.40 

23.60 

23.80 

24.00  24.20 

I 

e 

C.A 

0.H 

e 

LA 

Reduction  Bo  —  B  from  w-Table. 

-27 
-16 
-  5 
5 
16 
26 
35 
43 
50 
58 
67 
77 
88 

-20 
-10 
0 
10 
20 
30 
40 
50 
60 
70 
80 
90 
100 

-.50 
-.44 
-.38 
-.32 
-.26 
—  20 

+7.79 
.59 
.40 
.22 
7.05 
6  88 

7.86 
.   66 
.46 
.28 
7.11 
6  94 

7.93 

.73 
.53 
.34 
.17 
7  00 

8.00 
7.80 
.60 

.41 
.24 
7  07 

8.07 
7.86 
.66 
.47 
.30 
7  13 

8.14 
7.93 
.73 
.54 
.36 
19 

8.21 
7.99 
.79 
.60 

.42 
25 

8.28 
8.06 
7.86 
.67 
.49 
31 

8.35 
8.3 
7.93 
.73 
.55 
37 

8.42 
.20 
8.00 
7.80 
.61 
43 

.00 
-.01 
—  01 

.00 
.00 
00 

-.14 

-.01 

00 

.72 

.78 

6.84 

6.90 

6.96 

7.02 

7.08 

7.14 

.20 

.26 

-.08 
-.02 
+  04 

-.01 
-.01 
-  02 

.00 
.00 
00 

.57 
.43 
.29 

.63 
.49 
35 

.69 
.55 
41 

.75 
.60 
46 

.81 
.66 
52 

6.87 
.72 
58 

6.93 

.77 
63 

6.99 
.83 
68 

7.05 
6.89 
74 

7.11 
6.95 
,80 

+  .10 
+  .16 

+  .22 

-.03 
-.03 
-.03 

.00 
.00 
.00 

.16 
6.04 
5.91 

.22 
6.09 
5.97 

.28 
.15 
6.02 

.33 
.20 
6.07 

.39 
.26 
6.13 

.44 
.31 
6.18 

.49 
.36 
6.23 

.54 
.41 
6.28 

.60 
.47 
6.34 

.66 
.52 
6.39 

Date 

Jan. 

Feb. 

Mch. 

Apr. 

May 

June 

July 

Aug. 

Sept. 

Oct. 

Nov. 

Dec. 

5 

28.9 

30.6 

37.0 

44.2 

52.7 

62.0 

67.4 

67.0 

62.0 

53.2 

41.7 

33.1 

15 

27.9 

32.0 

39.5 

46.6 

55.7 

65.1 

68.5 

66.3 

59.9 

49.8 

37.6 

30.8 

25 

29.3 

34.5 

41.9 

49.6 

58.8 

66.2 

67.8 

64.2 

56.5 

45.7 

35.3 

29.8 

Note. — A  A  and  C.&O.H  have  been  united. 

II.— Reduction  to  the  3,500-foot  plane 


Temp. 

Correction  for 

22.40 

22.60 

22.80 

23.00 

23.20 

23.40 

23.60 

23.80 

24.00 

24.20 

I 

0i 

C.A 
0.H 

e 

AA 

Reduction  Bi  —  B  from  w-Table 

-24 

-20 

-.27 

+3.61 

3.65 

3.68 

3.71 

3.75 

3.78 

3.81 

3.84 

3.87 

3.90 

—14 

—10 

—  24 

.52 

56 

59 

62 

66 

69 

72 

75 

78 

81 

-  4 

0 

-.21 

.44 

.47 

.50 

.53 

.57 

.60 

.63 

.66 

.69 

.72 

fi 

10 

—  18 

36 

39 

42 

45 

48 

51 

54 

57 

60 

63 

16 

?0 

-.15 

.29 

.32 

35 

38 

41 

44 

46 

49 

52 

55 

?6 

30 

-.12 

22 

25 

28 

30 

33 

36 

39 

42 

45 

47 

36 

40 

-.09 

.  .  . 

.15 

.18 

.21 

.23 

.26 

.29 

.32 

.35 

.38 

.40 

45 

50 

-.06 

.08 

.11 

.14 

.16 

.19 

.22 

.25 

.28 

.31 

.33 

55 

60 

-.03 

.66 

.00 

3.02 

3.05 

.08 

.10 

.13 

.16 

.18 

.21 

.24 

.26 

64 

70 

.00 

-.01 

.00 

2.96 

2.99 

3.02 

3.04 

.07 

.10 

.12 

.15 

.17 

.19 

74 

80 

+  .03 

-.01 

.00 

.90 

.93 

2.96 

2.98 

3.01 

3.04 

.06 

.09 

.11 

.13 

84 

90 

+  .06 

-.02 

.00 

.84 

.87 

.90 

.92 

2.95 

2.98 

3.00 

3.03 

3.05 

.07 

94 

100 

+  .09 

-.02 

.00 

2.79 

2.82 

2.84 

2.86 

2.89 

2.92 

2.94 

2.97 

2.99 

3.01 

EXAMPLES    OF   BAROMETRIC   REDUCTION 

III.— Reduction  to  the  10,000-foot  plane 


49 


Temp. 

Correction  for 

22.40 

22.60 

22.80 

23.00 

23.20!  23.  40 

23.60 

23.80 

24.00 

24.20 

t 

02 

C.A 

e.H 

e 

AA 

'  Reduction  Bt  -  B  from  m-Table. 

-20 

-?0 

-.21 

-2.67 

2.70 

2.73 

2.75 

2.78 

2.80 

2.82 

2.85 

2.87 

2.89 

-10 

-10 

-.18 

-    .61 

.64 

.67 

.69 

.72 

.74 

.76 

.79 

.81 

.83 

1 

0 

-.15      ... 

-    .56 

.59 

.61 

.63 

.66 

.68 

.70 

.73 

.75 

.77 

12 

10 

-  12!      ... 

—    .51 

.54 

.56 

.58 

.61 

.63 

.65 

.67 

.69 

.71 

23 

20 

—  09 

—     46 

.49 

.51 

.53 

.56 

.58 

.60 

.62 

.64 

.66 

34 

30 

-.06 

-    .41 

.44 

.46 

.48 

.51 

.53 

.55 

.57 

.59 

.61 

45 

40 

-.03 

.00 

.66 

-    .37 

.39 

.41 

.43 

.46 

.48 

.50 

.52 

.54 

.56 

56 

50 

-.01 

+  .01 

.00 

-    .32 

.34 

.36 

.38 

.41 

.43 

.45 

.47 

.49 

.51 

67 

60 

+  .01 

+  .01 

.00 

-    .28 

.30 

.32 

.34 

.37 

.39 

.41 

.43 

.45 

.47 

78 
89 

70  +.03 
80  +.06 

+  .01 
+  .02 

.00 
.00 

-    .24 
-   .20 

.26 
.22 

.28 
.24 

.30 
'  .26 

.32 
.28 

.34 
.30 

.36 
.32 

.38 
.34 

.40 
.36 

.42 
.38 

100 

90 

+  .09 

+  .02 

.00 

-2.16 

2.18 

2.20 

2.22 

2.24 

2.26 

2.28 

2.30 

2.32 

2.34 

value  of  C  for  Santa  Fe  happens  to  be  0.00086  and  H  is  taken  in 
units  of  1,000  feet,  7.01.     Hence,  for  the  sea  level, 

for  0  =   -    20,  C.A0.H  =  0.00086  X  (-83)     X  7.01 

(-63) 


0 
20 
40 
60 
80 
100 


(-43) 
(-23) 
(-  3) 
(+17) 
(+37) 


-  0.50 
-0.38 
-0.26 

-  0.14 
-0.02 
+  0.10 
+  0.22 


The  plateau  stations  always  seem  to  require  such  a  correction 
in  order  to  make  a  harmonious  network  of  pressures  with  the 
surrounding  low-level  stations.  It  is  easier  to  make  this  correc- 
tion in  the  form  given  above,  rather  than  attempt  to  trace  out 
its  effect  upon  the  mean  temperature  6,  as  related  to  the  surface 
temperature  t.  The  entire  subject  needs  a  fuller  theoretical 
discussion  if  possible.  The  vapor  pressure  correction  e  is  the 
mean  value  as  for  the  argument  surface  /,  and  suffices  for  these 
station  reduction  tables  up  to  the  0.01  inch.  The  final  station 
reductions  to  the  sea  level  were  made  for  the  arguments  (t.  B}, 
and  applied  to  the  assumed  values  of  B,  so  that  for  the  same 
arguments  (t.  B),  the  value  of  BQ  is  immediately  read  by  an 
easy  interpolation. 

Similar  reductions  were  made  for  the  3,500-foot  plane,  and 
the  10,000-foot  plane.  They  were  checked  by  reduction  from  the 
station  to  3,500  feet,  to  sea  level,  from  sea  level  to  10,000  feet,  and 
thence  in  a  circuit  back  to  the  station  pressure.  This  was  done 
for  all  the  numerous  plateau  stations  in  the  United  States. 


CHAPTER  II 
Thermodynamic  Meteorology 

General  Formulas  for  the  Computation  of  P,  p,  R,from  the  Observed 
Temperatures  T  in  a  Free  Non-Adiabatic  Atmosphere 

IT  is  easily  seen  from  the  discussion  of  the  barometer  how 
many  complexities  this  instrument  introduces  in  practice,  on 
account  of  the  series  of  corrections,  and  by  reason  of  the  system 
of  units  employed,  which  separates  the  data  from  all  other 
thermodynamic  terms  occurring  in  meteorology.  There  is 
need,  then,  of  developing  another  system  of  reduction,  by  which 
it  may  be  possible  to  pass  from  the  temperatures  Tt  observed 
in  the  free  air  up  to  great  heights,  to  the  corresponding  pressures 
P,  densities  p,  and  gas  coefficients  R,  so  that  the  general  law 
P  =  p  R  T  shall  continuously  be  satisfied  throughout  the 
atmosphere.  If  the  mercurial  barometer  is  needed  on  the  surface 
to  give  a  base  for  vertical  reductions,  it  is  not  practical  to  carry 
it  to  heights  on  kites  and  balloons.  The  aneroid  may  be  used 
to  check  the  resulting  computed  pressures,  but  not  to  give  the 
actual  pressure  for  the  dependent  formulas.  Fortunately,  there 
is  a  simple  and  comprehensive  set  of  formulas  for  this  purpose, 
which  will  now  be  developed. 

For  any  temperature  vertical  gradient  a,  the  temperature 
T  at  the  height  z  above  T0  is, 

J   rr* 

(169)  T  =  TQ  -  a  z,  so  that,  d  T  =  -adz,  and  d  z  =  ---  . 

The  differential  equation  for  pressure  variations  with  the 
height  is  from  (41), 

(170)  —  dP  =  pg0dz=  --  p  godT,  by  substituting  d  z. 
From  the  Boyle-Gay  Lussac  Law,  P  =  p  RQ  T,  by  division, 


we  obtain,  since  by  (24)  —  ^- 


, 
a  /CQ       K  —  1 

dT       nk  dT 


aR0  T     ~  k  -  1   T 
50 


GENERAL   FORMULAS    FOR    NON-ADIABATIC    ATMOSPHERE       51 

Passing  to  logarithms  and  to  limits,  this  gives, 
-1  -  (^\ 

PO     \TO) 


nk 


Having  observed  7\  and  T0  on  two  levels,  at  the  vertical 
distance  apart  z\  —  z0,  the  pressure  PI  can  be  computed  from 
the  pressure  P0  on  the  lower  level.  We  proceed  to  determine 
the  density  pi,  and  gas  coefficient  RI,  which  correspond  with 
PI  =  piRiTi  on  the  Zi-level,  when  P0  =  p0RQTo  is  given  on 
the_2g-leyel.  By  successive  stages  from  the  surface  the  same 
formulas  will  arrive  at  any  altitude  where  the  temperature  TV- 
is  known.  From  two  successive  levels,  we  have  the  ratio, 

(173)      75-  =  —  5~^F  ,  and  by  transforming,  /^ 

r'o  p0  KQ  1  o 


(     . 
•   *; 

At  this  point  the  entire  treatment  of  thermodynamic 
meteorology  diverges.  If  the  gas  coefficient  is  taken  constant, 
RI  =  RQ,  and 

(175)      fl. 


For  example,  V.  Bjerknejs  in  equation  £Jj£ageJ>l,  "  Dynamic 
Meteorology  and  Hydrography,"  Carnegie  Institution  of  Wash- 
ington, 1910,  uses  this  form  for  the  ratio  pi/p0,  since  in  his 
system  of  units  go  =  1.  This  is  the  common  way  of  treating 
the  matter,  but  it  is  easy  to  see  that  this  derivation  of  the  non- 
adiabatic  densities  from  the  well-known  adiabatic  equation  is 
inconsistent  with  the  analogue  of  the  pressures  in  (172),  which 

k 
simply  multiplies  the  exponent  T  -  -  by  n,   so   that   for   the 

K  —  1 

densities  the  exponent  should  be 


,  _    . 

Proceeding  in  the  second  way  it  is  obvious  that,  preserving 
the  same  treatment  for  density  as  for  pressure,  we  should  take, 


52  THERMODYNAMIC   METEOROLOGY 


/17Av 

(176)      —  =  (  jT 
po         VI  o 


Ri       /TV*-1' 
Ro  =:  \T0) 

In  order  to  check  these  results  by  (172)  and  (173), 

-          ^    +(n-l)  +  l 

> 

which  is  correct.  This  process  makes  R  a  variable  in  the  existing 
non-adiabatic  atmosphere,  so  that  the  air  is  not  distributed  by 
gravitation  like  an  adiabatically  expanding  gas,  in  which  there 
is  no  circulation  and  no  change  of  heat  contents  by  radiation  and 
absorption  from  level  to  level.  On  the  contrary,  the  observations 
prove  that  usually  there  is  circulation  and  radiation  going  on 
to  preserve  the  gravitation  equilibrium  with  the  existing  pressure 
variations  or  gradients.  As  stated,  the  entire  system  of  ther- 
modynamics takes  on  a  new  form  through  the  fact  that  the 
specific  heat  must  also  be  a  variable  along  with  the  gas  coefficient. 

(2)         Cp-^R. 

We  shall  return  to  explain  the  consequences  of  this  funda- 
mental property  of  the  atmosphere,  which  is  in  reality  a  gaseous 
mixture  of  rapidly  varying  thermodynamic  capacities,  in  con- 
sequence of  the  effect  of  the  absorption  of  solar  radiation  and 
the  emission  of  atmospheric  radiation  in  various  ways. 

The  Adiabatic  Equations 

The  correlative  adiabatic  equations  follow  at  once  by  putting 
n  =  1,  and  a  =  a0, 


(179)  Pressure       = 

-TO         \J-  o 


(180)  Density^  =  (^ 

Po         W  o> 

(181)  Gas  constant  RI  =  R0. 


WORKING  NON-ADIABATIC  EQUATIONS  53 

The  Working  N on- Adiabatic  Equations 

(182)  Pressure,  log  P,  -  log  P0  =  —^  (log  7\  -  log  r0). 

(183)  Density,  log  Pl  -  log  Po  =  ^~  (log  7\  -  log  T0). 

(184)  Gas  coefficient,  log  RI  -  log  RQ  =  (»-!)  (log  Z\  -  log  T0). 

These  equations  were  published  in  the  Monthly  Weather 
Review,  March,  1906,  (38),  (39),  (40),  and  they  have  been 
illustrated  by  numerous  applications  to  balloon  and  kite  ascen- 
sions with  excellent  results,  up  to  great  altitudes,  as  20,000 
meters.  The  following  example  shows  the  method  of  arranging 
the  computation  so  as  to  proceed  from  level  to  level,  the  com- 
puted PI,  pi,  RI  of  one  becoming  P0,  p0,  R0  for  that  next  above 
it.  The  example  is  taken  at  random  from  our  computations, 
some  of  the  results  being  compiled  in  Bulletin  No.  3,  Argentine 
Meteorological  Office,  1913.  The  constants  are  taken  from 
Table  3  in  the  (M.  K.  S.)  system.  The  surface  values  of.P0,  PO, 
R0,  TQ  are  assumed  to  conform  to  the  adiabatic  system,  while  PI, 
pi,  RI,  TI  above  the  surface  are  computed  by  the  non-adiabatic 
system. 

At  the  height  z  =  116  meters  the  density  is  computed  from 
the  adiabatic  formula, 

P 


Also,  P  =  go  pm  Bo,  where  BQ  is  in  meters.  It  will  be  noted 
that  the  check  is  complete.  It  may  be  stated  that  the  ob- 
served values  of  B0  at  Lindenburg  are  usually  about  1  mm. 
higher  than  the  computed  values  Bc.  This  constitutes  a  cri- 
terion upon  the  adjustment  of  the  aneroid,  which,  in  ascending, 
lags  in  registration-  and  records  a  pressure  corresponding  to 
a  lower  level  than  that  assumed  for  the  temperature  T  at  the 
height  z. 


54 


THERMODYNAMIC   METEOROLOGY 


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WORKING  NON-ADIABATIC  EQUATIONS 


55 


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56 


THERMODYNAMIC   METEOROLOGY 


TABLE  15 

COMPARISON  OF  THE  PRESSURES  COMPUTED  BY  THE  NON-ADIABATIC 
FORMULA  AND  THOSE  OBSERVED  AT  GREAT  HEIGHTS 


Heights 

Lindenburg 

Lindenburg 

Atlantic  Ocean 

Atlantic  Ocean 

in 

May  5,  1909 

July  27,  1908 

Sept.  9,  1907 

June  19,  1906 

Meters 

Lat.  +52° 

Lat.  +52° 

Lat.  +26° 

Lat.  -2° 

z 

Be 

Bo 

Bc-B, 

Be 

Bo 

Bc-Bo 

Be 

B0 

Bc-Bo 

Be 

•Bo 

Bc-Bt 

19000 

18000 

.0589 

.0600 

-.0011 

17000 

.0679 

.0670 

+  .0009 

.0692 

.0690 

+    2 

.0742 

.0740 

+  !6662 

16000 

.0794 

.0790 

+    4 

.0808 

.0790 

+   18 

.0861 

.0860 

+    1 

15000 

.0927 

.0930 

3 

.0945 

.0950 

-    5 

.1004 

.1010 

-    6 

.1021 

.1020 

+  .0001 

14000 

.1082 

.1090 

-    8 

.1101 

.1110 

-    9 

.1177 

.1170 

+    7 

.1203 

.1200 

+    3 

13000 

.1266 

.1270 

4 

.1289 

.1300 

-   11 

.1381 

.1380 

+    1 

.1409 

.1410 

-    1 

12000 

.1484 

.1490 

6 

.1512 

.1520 

-    8 

.1615 

.1610 

+    5 

.1641 

.1640 

+    1 

11000 

.1744 

.1750 

6 

.1772 

.1780 

8 

.1881 

.1880 

-1-    1 

.1902 

.1900 

+    2 

10000 

.2041 

.2050 

-    9 

.2067 

.2080 

-   13 

.2179 

.2180 

-    1 

.2191 

.2190 

+    1 

9000 

.2374 

.2390 

-   16 

.2399 

.2410 

-   11 

.2509 

.2510 

1 

.2512 

.2510 

+    2 

8000 

.2746 

.2760 

-   14 

.2769 

.2780 

-   11 

.2875 

.2880 

5 

.2870 

.2870 

0 

7000 

.3159 

.3180 

-   21 

.3181 

.3190 

-    9 

.3281 

.3290 

9 

.3268 

.3260 

+    8 

6000 

.3619 

.3630 

-   11 

.3639 

.3650 

-   11 

.3732 

.3740 

8 

.3709 

.3710 

-    1 

5000 

.4128 

.4130 

-    2 

.4149 

.4160 

-   11 

.4233 

.4240 

-    7 

.4200 

.4200 

0 

4000 

.4696 

.4700 

4 

.4716 

.4720 

-    4 

.4789 

.4800 

-   11 

.4747 

.4750 

3 

3000 

.5330 

.5340 

-   10 

.5344 

.5340 

+    4 

.5402 

.5410 

-    8 

.5356 

.5350 

+    6 

2000 

.6035 

.6040 

-    5 

.6040 

.6040 

0 

.6077 

.6060 

+   17 

.6031 

.6030 

+    1 

1000 

.6823 

.6840 

-   17 

.6806 

.6810 

-    4 

.6821 

.6820 

+    1 

.6783 

.6780 

+    3 

Surface 

.7599 

.7599 

0 

.7551 

.7551 

0 

.7640 

.7640 

0 

.7610 

.7610 

0 

The  differences  between  Bc  and  BQ  are  probably  due  to  an 
assignment  of  the  temperature  to  a  slightly  erroneous  height,* 
owing  to  the  movement  of  the  balloon  ahead  of  the  record  of 
the  barograph  and  thermograph,  which  requires  a  correction 
for  lag.  The  pressure  recorded  by  the  aneroid  is  for  the  mixture 
of  dry  air  and  aqueous  vapor,  so  that  by  (75)  for  the  same  height, 
where  T  =  T0  and  p  =  p0, 

(185)      P0  =  P  (l  -  0.378  J; 

Bo  =  B  (l  -  0.378  ^ 

where    P   or   B  is   the  dry  air  pressure,   and  P0  or  BQ   the 
pressure   in  the  mixture,  e  being   the  vapor  pressure  in   the 

*  The  heights  have  been  read  from  the  aneroid  record  and  are  not  corrected 
for  the  supposed  lag;  but  the  error  is  less  for  T  than  for  P  because  T  changes 
more  slowly  than  P. 


THE    VARIABLE   VALUES    OF   H  =  -~ 


57 


same  system  of  units.  The  connection  between  (182)  and 
(159)  is  such  that  they  can  easily  be  shown  to  be  identical, 
after  the  action  of  the  mercurial  barometer  has  been  made  to 
equal  that  of  an  aneroid. 

The  Variable  Values  of  n  =  - 

The  introduction  of  n  into  the  adiabatic  formulas  converts 
them  into  the  non-adiabatic  formulas,  and  at  the  same  time  adds 
circulation  and  radiation  to  a  static  atmosphere.  Hence,  by  (13), 

=    _  _Tl-T0  _    -(Ta  -  To) 

a 


(186) 


n  =  —  =  — 


marks  the  natural  transition  from  static  to  dynamic  and  ther- 
modynamic  meteorology.  It  is  important,  therefore,  to  under- 
stand the  full  significance  of  the  ratio  between  the  adiabatic 
and  the  non-adiabatic  temperature  gradients.  Since  (Ta  —  T0) 


TABLE  16 
EXAMPLES  OF  THE  VALUE  OF  n 


Height  z 

Lindenburg 
May  5,  1909 

Lindenburg 
July  27,  1908 

Atlantic  Ocean 
Sept.  9,  1907 

18000 

-  3.6553 

17000 

+1.0966 

-16.4489 

-  2.5972 

16000 

-5.4830 

-  6.1684 

-  0.9399 

15000 

-9.8694 

-49.3467 

-  2.9027 

14000 

-3.5248 

-  6.1684 

+  4.9347 

13000 

-1.7624 

-  2.5972 

+  1.9739 

12000 

+5.8055 

+  2.0999 

+  1.3337 

11000 

+  1.1749 

+  1.1611 

+  0.9676 

10000 

+1.0846 

+  1.2986 

+     .1611 

9000 

+1.1215 

+  1.1344 

+     .1611 

8000 

+1.1749 

+  1.1749 

+     .3901 

6000 

+1.3520 

+  1.6179 

+     .3901 

4000 

+  1.7016 

+  1.8980 

+     .2653 

2000 

+3.5248 

+  1.2986 

+  2.5972 

1000 

-4.9347 

+  12.3366 

-12.3367 

and  (Ti  —  TO)  are  usually  each  negative  with  an  increase  in 
elevation,  n  is  generally  a  positive  quantity,  but  it  becomes 


58  THERMODYNAMIC  METEOROLOGY 

negative  whenever  there  is  an  inversion  of  temperature,  or 
temperature  increase  with  the  height,  as  near  the  surface  of  the 
ground  in  the  early  morning,  or  in  the  isothermal  layer  at  great 
heights.  If  TI  =  Ta,  n  =  1,  and  the  gradient  is  adiabatic; 
if  TI  =  TQ,  n  =  °°  and  there  is  no  temperature  change  with 
the  height;  if  TI  >  TQ,  n  is  negative,  and  if  TI  is  only  a  little 
greater  than  TQ  for  the  change  in  elevation  z\  —  z0,  n  will  be 
a  large  negative  quantity.  Table  16  gives  a  few  examples  of  the 
values  of  n. 

Table  16  indicates  the  wide  range  through  which  n  passes 
in  practical  reductions,  and  it  is  easily  seen  how  valueless  the 
formulas  become  for  meteorological  discussions  where  n  is 
assumed  to  be  unity,  as  is  commonly  the  procedure.  Further- 
more, since  the  value  of  n  must  always  be  carried  to  the  fourth 
decimal  it  has  not  seemed  worth  while  to  construct  general 
reduction  tables,  because  they  would  be  very  extensive  or  re- 
quire complex  interpolation. 

The  Differentiation  of  (172) 

Since  n  is  a  variable  in  equation  (172),  we  proceed  to  differ- 
entiate it  for  P,  T,  n,  variables. 


(187)        log  =  log          .    Differentiate, 


nk         TT  T         k     dn 


4-1 

r.  +  logr»  • 


k 
Substitute  P  =  p  R  T  and  Cp  =  R 


, 

rv  J. 

(190)  —  =  n  Cp  d  T  +  —CpTlQgTdn. 

p  M. 

Take  the  integral  between  limits  for  the  mean  p™, 

(191)  Pl~P°  =  %  Cfco  (3Ti  -  To)  +  ^  CA,  r10  log  p 

PlO  M  2  o 


DIFFERENTIATION   OF    (172)  59 

By  (144),  (146),  (147),  using  the  mean  values  Cpw, 

(192)  Pl~oP°  =  *  Cpw  (7\  -  To)  +  (m  -  n0)  Cpa 

(Tt  -  To). 

=  »!  c#10  (T\  -  r0)  +  »i  c/>a  (ri  -  r0)  - 

WoC>0  (Ti-To) 

=  cfco  (ra  -  r0)  +  cpa  (Ta  -  TO)  -  cpa 

(Ta  -  To) 

=  Cpa  (Ta  -  To)  -  (Cpa  -  Cfco)  (Ta  -  T0). 

Since  p  and  Cp  are  variable  in  the  stratum  (zi  —  z0),  the 
mean  values  are  pio  and  CpW)  while  HI  continues  constant 
within  the  stratum,  which  must  not  be  taken  too  thick  to  allow 
this  approximation  to  hold  true.  The  result  is  twofold.  First, 

(193)  Pl~P°  =  C#w  (Ta  -  To)  =  «i  C#i0  (rx  -  To). 

Pio 

The  adiabatic  system,  on  the  other  hand,  gives, 

(194)  Pra~P°  =  C#B  (rfl  -  To)  =  -  g  fe  -  so). 

PaO 

Hence,  the  difference  between  the  two  systems  is, 


(195)  -  -  =  (Cpa  -  Cp»)  (Ta  -  To). 
PaO                         PIO 

From  the  common  dynamic  equation  for  pressure  and 
velocity,  which  will  be  proved  in  a  later  section,  and  adding  a 
term  for  the  dynamic  energy  of  radiation  heat,  we  obtain  by 
substitution  in  (192)  the  working  and  fundamental  equation,  as 
the  second  result, 

(196)  go  fe  -  a.)  =  -  fl~P°  -  J  (?,2  -  <?o2)  -  «2i  -  Co). 

Pio 

These  have  already  been  quoted  in  (21),  (25),  (26),  (27). 
From  (192),  using  the  last  form,  we  find, 

(197)  go  (z,  -  Sb)  *  -  ^—  ^°  -  (Cpa  -  Cp10)  (Ta  -  To), 

Pio 

and  by  comparison  of  the  last  terms  (196),  (197), 

(198)  -  (ft  -  Co)  -  i  (?i2  -  <?o>)  -  -  (Cpa-Cp1Q)  (Ta-TQ). 


60  THERMODYNAMIC   METEOROLOGY 


(199)  -  (<2i  -  <2o)  =   -  (Cpa  ~  C#w)  (Ta-  To)  +  i  (?i2-?o2). 

That  is  to  say,  the  variation  from  the  true  adiabatic  system 
is  due  to  the  kinetic  energy  of  heat  (<2i  —  Q0),  and  circulation 
J  (qiz  —  g02)  for  the  unit  mass,  which  is  equivalent  to  the 
.variation  of  the  mean  specific  heat  of  the  stratum  from  the 
adiabatic  specific  heat,  Cpa  =  993.5787  in  Table  3,  multiplied 
by  the  change  in  temperature  between  the  bottom  T0  and  top 
Ta  of  the  layer  z\  —  ZQ.  Unfortunately,  there  seems  to  be  no 
way  to  separate  (Qi  —  QQ)  from  \  (qj  —  <?02)  through  the 
specific  heat,  except  by  using  the  direct  observations  of  the 
velocity,  and  then  computing  (Qi  —  QQ)  by  means  of  (199). 
Those  observatories  which  record  P,  T,  the  pressure  and  the 
temperature,  but  not  the  R.  H.  and  <?,  the  relative  humidity  and 
velocity  of  motion  of  the  stratum,  cannot  enter  upon  any  problem 
in  circulation  or  radiation  in  the  atmosphere. 

From  equation  (196)  we  obtain, 

(200)  -  —  =  gdz  +  qdq  +  dQ,  and, 

P 

(201)  -dP  =  gpdz+pqdq+pdQ. 
From  the  equation  (190)  is  derived, 

(202)  -  d  P  =  —  p  n  Cp  d  T  -  p  -^  Cp  T  log  T.  d  n, 

and  by  means  of  (146),  (144),  (21),  this  becomes, 

(203)  -  d  P  =  g  p  d  z  -  p  j£  Cp  T  log  T.  d  n. 

The  usual  adiabatic  pressure  variation  —  dP  =  gpdz,  as 
in  (170),  is  converted  into  the  non-adiabatic  form  with  circulation 

and  radiation  by  subtracting  p  T>  Cp  T  log  T  d  n,  which  includes 

the  differentials  first  for  heat,  and  second  for  circulation. 

If  equation  (203)  is  treated  in  the  same  manner  as  (170)  we 
shall  obtain  by  substitutions, 

foriA\    (T\  (T\  \   Zj-Zp  l      dp    Zi- 

(204)  f^)     -(~r)  =(a0-a)      T      = 


0  0 

which  is  the  difference  between  the  temperature  ratios  in  the 
adiabatic  and  the  non-adiabatic  systems. 


DIFFERENTIATION   OF    (172) 


61 


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THERMODYNAMIC   METEOROLOGY 


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DIFFERENTIATION    OF     (172)  63 

There  are  a  few  critical  remarks  that  can  now  be  made  to 
advantage,  in  view  of  the  principles  that  have  been  assumed  in 
many  important  meteorological  papers. 

(1)  By  (201)  and  (203)  it  is  not  proper  to  assume  that  the 
variation  of  the  pressure  in  a  vertical  direction  is  proportional 
to  the  mass  variation,    —  d  P  =  g  d  m,  because  this  excludes 
the  circulation  and  the  radiation.     It  is  a  contradiction  in  terms 
to  seek  for  solutions  of  radiation  problems  under  this  limitation. 

(2)  By  (198),   (199),  if  it    is  assumed  that  R  and  Cp  are 
constants  and  the  change  of  temperature  is  adiabatic,  there  can 
be  no  circulation  and  no  radiation.     It  is  a  contradiction  in  terms 
to  seek  for  solutions  of  radiation  problems  under  this  assumption. 

i*d  P 

(3)  By  (193)  the  integral  /  —  depends    upon   a  varying 

Cp.  It  follows  that  if  P.  T.  are  observed  by  the  instruments, 
if  RQ  is  taken  constant  in  the  Boyle-Gay  Lussac  Law,  and  if  p 
is  then  computed  directly  from  P  =  p  RQ  T,  this  value  in  the 

/dP 
—   will  always  lead  to  fictitious  results,  because  the 
p 

circulation  and  the  radiation  are  excluded  from  the  dis- 
cussion. 

(4)  In  the  integration  of  (190),  in  strata  where  n  is  constant, 
as  they  can  be  made  by  taking  the  layer  thin  enough,  it  is  proper 

to  use  the  mean  values  of  the  several  terms,  pio,  Cpio,  TIQ, 

jp       p  _  _  p 

1  i  /  "\ 

PlO    =    2    \Pl  +   POJ. 


/*££ 
J    P 


f 


Pio 
nCpdT  =  n  Cp1Q  (^  -  T0),  Cpw  =  J  (Cpi  +  Cp0). 


I  T  I  TI 

M  TQ  M  TQ 

since  Cp  =  Cpa  and  n0  =  1  in  this  term. 

(5)  It  has  been  customary  to  evaluate  the  equations  of 
motion  in  the  atmosphere  while  omitting  the  heat  term—  (Qi— Qo) 
in  (196).  The  result  is  that  it  is  impossible  to  balance  the  other 
three  terms,  so  that  the  great  problem  of  the  relation  of  the 
circulation  to  the  observed  pressures  in  the  general  and  the  local 
circulations  has  been  insoluble.  In  order  to  exhibit  more  fully 


64 


THERMODYNAMIC   METEOROLOGY 


the  relation  of  these  four  terms,  two  examples  of  balloon  ascen- 
sions to  great  heights  are  added. 

TABLE  18 
THE  EVALUATION  OF  —  (Ql  —  Q0)  IN  BALLOON  ASCENSIONS  (196) 


Station 

Lindenburg,  May  5,  1909 

Atlantic  Ocean,  Sept.  9,  1907 

jj 

* 
_/_      ._  \ 

Pi-Po 

-J 

-(Qi- 

Pt-Po 

-J 

_(Ql_ 

pio 

(v-^ 

Qo) 

pio- 

(V-tf 

Qo) 

17000 

9806.0 

6437.6 

+  3.6 

3358.6 

+  6.2 

6755.8 

-26.8 

3065.9 

+11.1 

16000 

9806.0 

6629.2 

+  5.6 

3169.2 

+  2.0 

7301.2 

+  6.4 

2496.1 

+2.3 

15000 

9806.0 

6963.7 

+11.2 

2817.1 

+14.0 

7886.4 

+38.9 

1885.6 

-  4.9 

14000 

9806.0 

7357.0 

+  9.2 

2433  .  5 

+  6.3 

8288.6 

+104.8 

1424.2 

-11.6 

13000 

9806.0 

7851.0 

+  5.6 

1939.7 

+  9.7 

8508.6 

+53.0 

1228.5 

+15.9 

12000 

9806.0 

8300  .  4 

+  1.7 

1499.8 

+  4.1 

8666.2 

-89.3 

1227  .  9 

+  1.2 

11000 

9806.0 

8489  .  0 

-  0.9 

1313.9 

+  4.0 

8706.6 

-33.2 

1130.2 

+  2.4 

10000 

9806.0 

8530.6 

+  0.9 

1268.8 

+  5.7 

8721.0 

-   1.4 

1079.9 

+  6.5 

9000 

9806.0 

8565.0 

+  8.1 

1227.5 

+  5.4 

8769.4 

+  4.2 

1025.7 

+  6.7 

8000 

9806.0 

8615.8 

+13.2 

1176.9 

+  0.1 

8845.2 

+30.4 

928.1 

+  2.3 

7000 

9806.0 

8668.8 

+16.1 

1114.1 

+  7.0 

8958.0 

-54.3 

895.0 

+  7.3 

6000 

9806.0 

8750.2 

+  8.8 

1044.0 

+  3.0 

9074.0 

-53.8 

777.6 

+  8.2 

5000 

9806.0 

8885.0 

+  5.2 

909.2 

+  6.6 

9222  .  0 

+18.3 

568.1 

-  2.4 

4000 

9806.0 

9051  .  5 

+  5.4 

745.6 

+  3.5 

9340.5 

-24.6 

485.9 

+  4.2 

3000 

4903.0 

4589.3 

+  7.0 

312.3 

-  5.6 

4706.0 

-  2.2 

204.1 

-  4.9 

2500 

4903.0 

4636.4 

+11.7 

257.5 

-  2.6 

4725.8 

-  6.2 

179.2 

+  4.2 

2000 

4903.0 

4689.1 

-  5.9 

217.3 

+  2.5 

4764  .  9 

-  4.5 

135.7 

+  6.9 

1500 

4903.0 

4738.9 

-22.1 

174.4 

+11.8 

4815.6 

-  0.8 

87.6 

+  0.6 

1000 

4903.0 

4813.8 

-32.5 

104.4 

+17.3 

4846.6 

+13.7 

9.9 

+32.8 

500 

3765.5 

3753.8 

-11.5 

19.5 

+  3.7 

4865.7 

-10.7 

0.9 

-  1.9 

*  More  exactly  g  should  diminish  with  height. 

It  is  easy  to  see  that  in  the  higher  levels  (Qi  —  Qo)  is  a  domi- 
nant term,  and  that  there  is  no  possibility  of  a  balance  between 
the  other  three  taken  by  themselves..  There  is  a  delicate 
interaction  of  the  four  terms,  such  that  the  circulation  seeks  to 
adjust  the  pressure  and  the  radiation  to  the  demands  of  gravity. 
The  minor  errors  A  are  to  be  chiefly  ascribed  to  the  fact  that 
the  observed  temperatures  are  not  quite  correct  at  the  height  z. 
In  many  cases  there  are  pair  values  of  -f  A  and  —  A  easily 
adjusted  on  this  hypothesis. 

The  Two  Laws  of  Thermodynamics 

Before  proceeding  further  it  is  necessary  to  summarize  the 
two  fundamental  laws  of  thermodynamics,  the  first  being  that 
of  the  conservation  of  energy,  and  the  second  being  that  of  the 


THE   TWO   LAWS    OF   THERMODYNAMICS  65 

decrease  of  entropy  or  increase  of  expenditure  in  a  non-conserva- 
tive system.  The  first  law  is  defined  as  follows:  In  a  conservative 
system,  from  which  no  heat  escapes  and  into  which  no  heat  is  re- 
ceived, the  sum  of  all  the  changes  in  the  energy,  whether  large  or 
small,  remains  constant.  In  a  system  in  communication  with  the 
outside  world  the  amount  of  the  energy  gained  or  lost  by  it  is 
equal  to  that  delivered  to  or  received  from  the  outside  world. 
The  second  law  may  be  described  in  several  ways.  Heat  cannot 
by  itself  pass  out  of  a  colder  into  a  warmer  body  (Clausius). 
It  is  impossible  to  construct  a  periodically  acting  machine  which 
does  nothing  else  than  raise  a  weight  in  expending  work  and 
cooling  a  reservoir  of  heat  (Planck).  There  exists  in  nature  a 
quantity  which  always  changes  itself  only  in  the  same  direction 
in  all  the  variations  which  take  place  in  nature;  this  is  the 
entropy.  It  is  in  nowise  possible  to  diminish  the  entropy  of  a 
system  of  bodies  without  there  being  left  changes  in  other  bodies. 
If  such  changes  do  not  remain,  then  the  entropy  of  a  system  can 
continue  the  same.  Every  physical  or  chemical  change  takes 
place  in  such  a  way  that  entropy  either  decreases  or  remains 
the  same,  but  the  outside  world  tends  continually  toward 
maximum  entropy.  It  is  necessary  and  sufficient  for  the 
equilibrium  of  a  separate  structure  that  for  all  possible  changes 
in  the  state  of  the  structure,  the  changes  of  entropy  be  zero  or 
negative  (Weinstein). 

The  First  Law  of  Thermodynamics 

The  following  series  of  formulas  are  available  in  the  adi- 
abatic  and  non-adiabatic  systems  using  the  proper  values  of 
P.  p.  R.  T. 

P 
p  ~'~  R  f 

RT 

i)  =  — — . 


dT  dP 


(205) 
(206) 
(207) 
(208) 

P 
Pv 

dT 

d  v 

=  PRT 

=  RT. 
Pdv 

+  vd 

P 

dP 

=  P 

R 

R 

T 

dT        P    '    P 

66  THERMODYNAMIC   METEOROLOGY 

dv       dT       dP_ 

(209)  —  :     ~Y        p  • 

Referring  to  the  series  of  equations  84-118  for  the  definition 
of  the  terms,  we  have  generally, 

(210)  Inner  Energy.         U  =  H  +  J  =  Q-W. 

(211)  U  =  Cv  fT.pdv  =  -£  fp.dv. 

(212)  dU 
(213) 

7?  7"1 

(2H)  d  U  =  Cv  d  T  +  --  dv  -  Pdv. 

(215)  dU  =  CpdT  -  Pdv. 

(216)  d  U  =  Cvd  T  =  (Cp  -  R)  d  T. 

(217)  External  Work.       W  =  (K)  +  V  =  J  +  H  +  V. 

(218)  dW  =  dQ-dU. 

(219)  dW  =  TdS  -dU. 

(220)  dW  =  d(TS-  U}  -SdT. 

(221)  dW  =*  -dF  -SdT. 

(222)  d  W  =  P  d  v. 

dP 

(223)  dW  =  RdT  -—  . 

(224)  Heat  Energy.   Q  =  W  +  U. 

(225)  ()  =  TF  +  #  +  /  =  #  +  #- 

(226)  Q  =  [(K)  +  V]  external  +  [H  +  J]  inter- 

nal +  (R)  friction. 


(227) 

(228) 

(229  dQ  =  CvdT-\-RdT  —  v  dP  =  Cp  d  T  — 


(228)  JQ  =  Cz;jr+  ~  -  dv  =  Cvd  T  +  Pdv. 


THE  FIRST  LAW   OF   THERMODYNAMICS  67 

(230)  dQ-C, 


K 

(231)  dQ  =  TdS. 
From  (218)  in  heat  units,  we  obtain, 

(232)  First  Law.  A  d  W  =  d  Q  -  A  d  U,  and  by  (222), 

AdU 


(233)  A  dPd       ..±(AQ\         d 

[   dTdv     ~  dT\dv)    ~J~v 

dP       dAP       dCv 

(234)  A  IT  =  ^T  ~  ^ 


*(->) 

V     * 


<** 


(236)  -  Qi  =  -  Ui  -  Wi,  Heat  expended  outward  (negative). 
+  ft  =  +  #2  +  P^>,  Heat  received  inward  (positive). 

(237)  A  Q  =  Q2  -  Qi  =  (U2  -  Ui)  +  (W>  -  Wi),    Resultant 

heat  energy. 

Among  the  definitions  we  have, 

V  =  the  external  gravity  potential  acting  inwards,  to- 
gether with  the  centrifugal  force  of  the  earth's 

rotation  at  the  angular  velocity  w0  =   —  ,     at 
the  perpendicular  distance  tcr  from  the  axis, 

(238)  V  =  -  •  fg  d  z  +  y*f  (vutfdm. 

(239)  (K)  =  the  kinetic   energy  of  motion   of   the  mass  m 

with  the  velocity  q.     (K)  =  |  m  (f  =  H  +  /. 

(240)  F  =  the  free  energy  or  the  thermodynamic  potential 

at  a  constant  volume.        F  =  U  —  T  S. 

(241)  U  =  The  bound  energy.  U  -  F  =  T  S. 

(242)  0  =  the    thermodynamic    potential     at    a    constant 

pressure,  F  +  Pv.  <f>  =  U  -  TS  +  Pv. 

=  U-TS  +  RT. 


68  THERMODYNAMIC   METEOROLOGY 

Fundamental  Equations  and  Definitions 

It  is  convenient  to  have  for  ready  reference  the  fundamental 
equations  and  several  definitions  of  thermodynamic  processes, 
though  they  cannot  be  further  developed  in  this  connection. 
It  is  the  purpose  of  this  treatise  to  prepare  such  data  for  meteor- 
ology as  can  be  admitted  into  the  large  group  of  well-known 
equations  which  have  been  heretofore  inapplicable  in  the  atmos- 
phere for  lack  of  the  necessary  correct  values  of  P.  />.  R.  T. 
There  are  several  variables  :  P.v.T.  Q.W.U  .S.R. 


(243)        dU  =  AldQ-  dW  =  AlTdS  -  Pdv. 
=  Pdv. 


(244) 

dF  =  -  A^SdT  -  Pdv. 

f»A*\         JT  dF  +  Pdv  dF 

(245)         dT==~~~A^S  —  '      dv==~~ 

d<f>  =  - 

Adidbatic  Processes,    d  Q  =  0  and  d  S  =  0. 
Adiabatic.    d  Q  =  0  signifies  no  gain  of  heat  from  the  outside 
and  no  loss  of  heat  to  the  outside  from  the 
system. 
Isentropic.    dS  =  0  signifies   that   the   entropy  remains   con- 

stant. 

Isodynamic  Processes,    d  U  =  0  and  d  T  =  0. 
Isodynamic.  d  U  =  0.     The  inner  energy  remains  constant. 
Isothermal,    d  T  =  0.     The  temperature  remains  constant. 

Isoenergetic  Processes.     dW  =  0,  d  P  =  Q,  dv  =  0,  d  p  =  Q. 
Isometric,     d  W  =  0.     The  expenditure  of  external  work  is  the 

same. 

Isobaric.        d  P  =  0.     The  pressure  remains  unchanged. 
Isochoric.       dv  =  0.     The  volume    is    constant    during    the 

process. 

Isopyknic.      d  p  =  0.    The  density    is    constant    during  the 

process. 


FUNDAMENTAL  EQUATIONS   AND   DEFINITIONS  69 

Isopiestic  Processes,    d  P  =  0. 

Isopiestic.       dP  =  0.    The    pressure    is    constant   while  the 

other  variables  change. 

Isoelastic  Processes.    dR  =  0. 

Isoelastic.       dR  =  0.     In    ideal    gases    the  gas  coefficient  is 

constant. 

Evaluation  of  dQ  in  Terms  of  P.  v.  T.  through  the  Entropy  S. 
Taking  the  following  three  pairs  of  variables,  they  lead  to 
the  definition  of  the  specific,  latent,  and  expansion  heats. 


(246)     Variables  (v.T).    d  Q  =  T  (||)  dT  +  T  (~)    d  v  = 

\O  JL  '  v  VO  V  '  T 

+  CTdv. 


(247)  Variables  (P.  T).     dQ  =  T  d  T  +  TJ     dP 


(248)  Variables  (P.v).      d  Q  =  T  (|)  dT+T  (|)    d  v  = 


Hence  by  comparison  the  definitions  become, 
Specific  Heats.  Latent  Heats.          Expansion  Heats. 

(249)  CP  =  T  (  .  (250)  CT  =  r(  (251)  rp=  r(|f  )   . 


p.  T  p  p 

(252)  c,  =  T  .  (253)  rr  =  r.  (254)  r>  r) 


These  occur  in  These  occur  in          These  occur  in 

radiation.  evaporation.  convection. 

The  subscript  indicates  the  term  which  remains  constant. 

Evaluation  ofdQin  Terms  of  P.  v.  T.  through  the  Inner 
Energy  U. 


(255)  Variables 


(r.  T).  dQ  =A  (|^)  d  T  +  A  [  (|^)  T+P~]d  v. 


70 


THERM  ODYNAMIC  METEOROLOGY 


(256)  Variables  (P.  T).  d  Q  =A  [  (|~)   +  P  |~]d  T  + 


(257)  Variables  (P.  v)  .  d  Q  =  A  (|^)  d  P  +  f  (|^)    +  P~| 

*  o  Jr  '  p  U»AO  v  *  —i 

Specific  Heat 


d  v. 


p 

Latent  Heat 


Expansion  Heat 

^  -  () 


Latent  Heat 


Specific  Heat 

-  (H). 

-dfi 

Expansion  Heat 


(263)      r,= 


By  inter  comparisons  and  substitutions  very  numerous  equa- 
tions can  be  constructed.  Compare  Weinstein's  "Thermody- 
namik."  Those  for  entropy  are, 

(264)  SvT  =  S0  +  Cv  log  T  +  A  R  log  v. 

(265)  STP  =  S0  +  Cp  log  T  -  A  R  log  P. 

The  Second  Law  of  Thermodynamics 
This  is  derived  from  equation  (231). 

(266)  Second  Law.    dS  =-~. 

This  gives  rise  to  two  processes  in  nature,  the  reversible,  in 
which  after  a  series  of  transformations  the  original  state  is 
reached,  and  the  irreversible  process,  in  which  the  original  state 
is  permanently  lost. 


THE   REVERSIBLE    AND    IRREVERSIBLE   PROCESSES  71 

For  the  reversible  process,  d  S  =  -=r  =  0. 

From  (225)  for  d  Ui  =  d  U2  =  d  W2  =  0,  we  have, 

(267)  Q2  -  Ql  +  W1  =  0. 

(268)  From  the  first  law.        d  Q2  -  d  Qi  +  d  Wi  =  0. 


(269)  From  the  second  law.    d  S2  -  d  Si  =  -r  -  -       =  0. 

•L2  1\ 

T 

(270)  Solving  these  equations,    d  Qi  =      ^d  Wl' 


r,'    a  -  Q2  "  TI  -  2Y    d  -  g2  "  TX  -  r2* 

(272)  ^  =  (Tl~  ^  =1  -  ^-2.    The  efficiency  of  the  engine. 
^i  ^  i  •»  i 

^     of1       M      ^  -  n  (  1      ±-} 
-  ft       -   r.         --  ft       -       - 


(274)  Carnot's   Function  =  -^  in  mechanical  units  for  7\  >  T2 

1  1 

=  -T-TFT  in  heat  units. 
A  T! 

T!  =  the  temperature  of  the  source  of  heat  energy. 
T2  =  the  temperature  of  the  sink  of  the  energy. 
The  energy  runs  down  from  the  source  to  the  sink. 

For  the  irreversible  process,    d  S  =  -TJT  >  0. 

(275)  From  the  first  law.        d  Q2  -  d  Ql  +  d  Wi  =  0. 

(276)  From  the  second  law.    d  S2  -  d  Si  =  -J^-  -  -Jr*  >  0. 

lz  J-i 

Solving  these  equations. 

(277)  dWl 


72 


THERMODYNAMIC  METEOROLOGY 


(278)  Ford  IF,  =  0, 

(279)  d  Qi  >~r 


. 

1  1 


d  Q2 


Carnot's  Cyclic  Process 

As  an  example  of  a  reversible  process  we  may  describe  the 
Carnot  Cycle,  in  which  a  unit  mass^with  the  initial  condition 
(Pi.  »i.  jfi)  passes  by  an  isothermal  change  to  a  second  condition 


O  Volume  v 

FIG.  3.     Carnot's  cycle 

(P2-  »2.  ^i),  then  by  an  adiabatic  change  to  the  third  condition 
(Ps.  z;3.  T2),  thence  reversing  from  this  extreme  point  by  another 
isothermal  change  to  (P4.  z>4.  T2),  and  finally  by  another  adiabatic 
change  to  the  initial  (Pi.Vi.  TI).  This  is  illustrated  in  Fig.  3. 

Ti  =  temperature  of  the  source  RI. 
Tz  —  temperature  of  the  sink  R2. 

1  and  3  =  Isothermal  processes. 

2  and  4  =  Adiabatic  processes. 
1  and  2  =  Work  of  expansion. 

3  and  4  =  Work  of  compression. 
+  Q  =  Heat  received. 

—  Q  =  Heat  expended. 

+  W  =  Work  of  expansion. 

—  W  =  Work  of  compression. 


CARNOT'S  CYCLIC  PROCESS  73 

Summarizing  by  the  first  law. 

Isotherm.  Adiab.          Isotherm.  Adiab. 

(280)  Q  =  Ql-Q2  =  W 

=  Wi  +  W2  -W3  -Wt 

nTi  V3T*  vi  Ts  vtTi 

(281)  =    IPdv       +/Pdv      -IPdv  -IPdv, 

Jv\  Ti  Jv2  Ti  Jv*  T*  Jvi  Ti 

(282)  Since P  =  ^    =  -  Cv  j^- by  (228)  iordQ  =  0. 

/•Ti  rT*  rV4T2  rTl 

=  R  f   -dv  -  fCvdT  +R/    ~dv  -  fCv  d  T, 

JVI     v  JTI  Jvz     v  JT-L 

(283)  Since  R  log  v  +  Cv  log  T  =  Const,  by  (228)  for  d  Q  =  0. 


(284)  =  R  (T,  -  r.)  log  ^  =  R  (T,  -  T2)  log  = 

It  follows  that  log  —  =  log  — ,  or  —  =  — . 

(285)  (k  -  1)  log  Pl  =  log  TV     By  (180). 

(286)  +(k-i)  log  vi  +  log  Tt  =  (k-  1)  log  v,  +  log  TV 

(287)  +  (k  -  1)  log  v2  .+  log  T!  =  (k  -  1)  log  fl3  +  log  T2. 

From  the  Second  Law  by  (271)  and  (180), 
k- i  , k- i 


LATENT  HEAT 
Cyclic  Process  for  Vapors  at  Maximum  Pressure 

A  second  example  of  the  reversible  process  is  found  in  the 
cycle  through  which  vapors  pass  in  changing  to  liquids,  by  the 
latent  heat  which  is  required  in  effecting  this  transformation. 


74 


THERMODYNAMIC   METEOROLOGY 


Vapor  Liquid  Solid        Total 

Mass  Mi        M2  M3     M  =  Mi  +  M2  =  Constant. 

Specific  Heat    Ci         C2  C3 
Latent  Heat      ...         r\  r2 

Volume  Vi  v2  v3  v  =  Vi  +  v%. 


Liquid 


\ 


Vapor 


-Q, 


\ 


O          e  h  f  0    V 

FIG.  4.     Cyclic  process  for  maximum  vapor  pressure 

(289)  Product.  M  (vi  +  v2)  =  Mi  Vi+M2  v2  =  Mi  Vi  +  (M  —  MI)  v2. 

(291)  dMv  =  M dv  =  (vi  —  v2)  dMi. 

This  is  the  mass  which  evaporates  in  the  expansion  while 
(z>3  —  z;4)  d  M2  condenses  during  compression.     Hence 

(292)  ^  =  -^- 

d  v        Vi  —  v2 

The  general  equation  of  condition  is, 

(293)  A  P  d  v  =  r2  -  ,— •  d  v  =  r2  -      -  d  v  in  heat  units. 

d  v  Vi  —  v2 

For  d  v  =  0  we  have  by  (209),  P  =  T  -r^.    Hence, 

dP  M 

(294)  APdv  =  ATT^dv  =  r2—-dv,s 


(295) 
(296) 


r2 


A  Ti       ~       T       latent  heat  of  vaporization 


of  liquid  to  vapor  (^i  —  v2)  in  heat  units. 


r3  =  A  T2 


^-2-  7-=-,  latent  heat  of  melting  of 
i      a  1  2 

solid  to  liquid  (v2  —  v3)  in  heat  units. 


SECOND  FORM  OF  THE  EQUATIONS  FOR  LATENT  HEAT   75 

The  Second  Form  of  the  Equations  for  Latent  Heat 

For  external  equilibrium  where  there  is  no  exchange  with  the 
surrounding  medium,  the  conditions  are  : 

Total  Vapor      Liquid        Solid 

(297)  Masses.  M  =  M  i        +  M2       +  Ms. 

(298)  Specific  volumes,    v  =  v\          +  %          +  Vs- 

(299)  Volume.  Mv  =  M  i  Vi    +  M2  v2    +  M  3  %. 

(300)  Energy.  M  Z7  =  Mi  C/i  +  Jlf  2  U2  +  Ms  U3. 

(301)  Entropy.  MS  =  M  l  Si  +  M  2  S2  +  M  3  5*3. 

For  internal  equilibrium  the  general  equation  is, 

(302)  yl1  M  5  5  =  A1  S  Ma  5  5a  +  ^l1  S  Sa  8  Ma, 
Where  a  takes  the  values  1.  2.  3.  in  succession. 

Since  U  =  Q  -  W,  and  TS  =  U  +  W  =  U  +  Pdv  =  Q, 
we  have  by  differentiation  and  substitution, 

(303)  SS  =  ^  +  L^ 

(304)  A*M  SS  =  S  ^-^  +  2  Ma^3ga  +  A*Sa  SMa=  0. 

-^  a  -'a 

The    three    independent    conditions    for   interpreting    this 
equation  are, 

(305)  For  the  masses,        S  6  Ma  =  0. 

(306)  For  the  volumes,      S  Ma  5   va  +  2  »fl    5  Afa  =  0. 

(307)  For  the  energies,      S  Afa  6  £/a  +  S  Z7fl  6  Ma  =  0. 
After  eliminating  from  the  equations  6  Jl/2  .    5  V2  .    6  C/2- 

(308)  A^  d  S  =  (4-  -  7^-}  M,  d  Ui  -  (^  -  ^-}  M*  d  U3 

\1  1  1  2/  \^  2  ^  3/ 


=  0. 


76  THERMODYNAMIC  METEOROLOGY 

The  conditions  of  equilibrium  for  the  maximum  entropy  are 
uniform  temperature,  T  =  T\  =  T2  =  jT3,  and  uniform  pressure, 

Pi  =  P*  =  Pi. 
Hence  we  have  by  selection, 

(309)  r2  =  Ti  (Si  -  52)  =  A  (Ui  -  U2)  +  PI  (vi  -  v2)  for  vapor 

and  liquid. 

(310)  r,  =  T2  (S2  -  St)  =  A  (U2  -  Z78)  +  P*  (v*  -  %)  for  liquid 

and  solid. 

SPECIFIC  HEATS 

A  third  example  of  a  reversible  cyclic  process  is  given  in  the 
specific  heats,  C\  for  vapor,  C2  for  fluid.  From  the  first  law  by 
(234),  we  have, 

,   dP      d.AP      d.Cv 
(234)     A  jf  =  -^       ~j^. 

Develop  these  terms  successively  by  substitutions. 
Differentiate  the  last  form  of  (293),  dividing  by  d  v, 

d'AP          M 


r2  d  (DI  —  v2)  -i     M 
Ldf  ~  (vi  -  v2)  d  T]  vi-v* 

The  specific  heat  of  the  mixture  is,  by  (136), 
(312)  Cv  =  C2(M-M1)+CiMi-r2j~.     Hence,  by  (289), 


'fQiA\  f^\^L  -       C*dMi       CidMi  r2        \~dvi-d  ^2"1 

L4j  (2)  dv   "  dv  dv      "  (v!  +  v2)  I     dT      J 

^     By  (292), 
dCv 


Subtract  (1)  -  (2). 


SPECIFIC  HEATS  77 


/»i^  i     ^          /-  .  .     , 

(317)  -7-=;  -\-  C2  —  Ci  =  A  —  TT  —  -p=  in  heat  units. 
#  i  M.        u,  L 

The  Specific  Heats  in  Terms  of  the  Latent  Heats 

It  can  be  proved  by  differentiation  of  the  first  forms  of  (309) 
and  (310)  and  the  necessary  substitutions  that, 

(318)  (Cfc  -  (Q)2  -  f*  -  »  +  _*_  [(f|)p_  f  ?)  J, 

vapor-liquid. 

<™>  «*>•  -  «»>•  -I?  -  ?  +  ^[.(t?),-  (I?)  ,) 

liquid-solid. 
Compare  Planck's  "Thermodynamik." 

Examples  of  the  Thermodynamic  Data 

1.  Carnot's  Cycle.    In  Fig.  3  the  area  enclosed  between  the 
isotherms   (1.3)   and  the  adiabats   (2.4)   represents  the    work 
done,  W,  in  the  cyclic  process,  and  the  figure  is  called  the  indicator 
diagram.    This  is  used  in  studying   the  efficiency  of    engines, 
whether  the  process  is  natural  or  mechanical,  and  there  is  a  very 
large  literature  on  the  subject.     No  applications  have  as  yet 
been  made  in  the  atmosphere,  for  two  reasons,  the  first  because 
of  the  difficulty  of  tracing  out  the  history  of  a  given  mass,  and 
the  second  because  the  values  of    Cv  in  the  formula  are  not 
constant,  and  the  true  values  of  it  have  not  heretofore  been 
computed.    However,  it  is  possible  to  take  a  standard  mass, 
as  one  kilogram  of  air  at  an  initial  point,  and  trace  out  the 
conditions  through  which  it  must  have  passed  in  rising  from  the 
surface  through  changing  P.  v.  T.  till  it  arrives  at  the  surface 
again  in  the  original  state,  even  though  the  path  during  its 
circulation  may  not  be  known.    This  work  will  be  reserved  for 
further  studies. 

2.  Cyclic  Process  for  Vapors.    In  Fig.  4,  we  have, 

(320)  (z>2  —  Vi)  d  Mi  the  mass  which  evaporates  in  expansion. 

(321)  (z>3  —  z>4)  d  M2  the  mass  which  condenses  in  compression. 


78  THERMODYNAMIC   METEOROLOGY 

(322)  b  k  =  -TJ,  d  T  the  increase  in  the  pressure. 

(323)  W  =  (vi  —  v2)  -T-,  .d  Mid  T  the  work  done  in  the  area 

a,  b,  c,  d. 

(324)  Qi  =  r2dMi  =  C2d  MidT  the  heat  received  in  expansion. 

(325)  Q2 


(326)    Q2  =  CidMidT  --.d  Mi  d  T  the  heat  expended  in 


compression. 

(327)  Q  =  Qi-Q2=  (j^  +  C2- 
Since  A  =  ^,  this  becomes  as  in  (317), 

(328)  -~  +  C2  -  Ci  =  A  (vi  -  v2)  jj,  in  heat  units. 

Values  of  the  Latent  Heat  and  Specific  Heats 

(329)  r2  =  606.5  -  0.708  t  for  water  to  aqueous  vapor. 
n  =  80.066  for  ice  to  water. 

Cpi  =  0.4810  Cp2  =  1.0000  Cp*  =  0.5020. 

Example  3.     Water  to  Vapor  at  100°  C.,  by  Vaporization 

Ti  =  273  +  100  =  373. 

z>i  =  1658  the  volume  of  1  gram  of  aqueous  vapor  at  100°  C. 

^2  =  1  the  volume  of  1  gram  of  water  at  0°. 

vi-v2  =  1658  -  1  =  1657. 

-r~  =  27.2  millimeters  of  mercury. 

-  2-3894  x  10~8  =  H85TOOO-  ^  =  2'37829  -  10) 


-  X  =  3.1856  X  10  ~5     (log  =  5.50319  -  10) 

go  7oU 

reduction   from    work    units    and  mm.    to    C. 
G.  S.  units  heat. 


VALUES  OF  LATENT  HEAT  AND  SPECIFIC  HEATS      79 

(295)  Latent  Heat.  r2  =  —  T:  fa  -  %)  ~  =  373  X  1657  X 

go  a  1 

27.2  X  3.1856  X  10  "5  =  535.5. 

Example  4.    Ice  to  Water  at  0°  by  Melting 
T2  =  273  +  0  =  273. 
vz  =  1.00  volume  of  1  gram  of  water  at  0°. 
v3  =  1.09  volume  of  1  gram  of  ice  at  0°. 
%  -  v3  =  -  0.09. 

d  P 
—  -r=  =  134.6  atmospheres. 

(296)  Latent  Heat.   r8  =  — .  T2  fa  -  vs)  ~  =  273  X  (-  0.09) 


(-134.6)  X  3.1856  X  10  ~5   X  760  =  80.066. 
Example  5.    Aqueous  Vapor  in  Contact  with  Water  at  100°. 

&--  0.708          (||)     =  4.931          (||)     =  0.001 
dT  \oTJ  p  \o  1  J  P 

Cp2  =  1.0300  P  =  760mm.      T  =  373 

<3,8,    », 


0.001]  =  -  0.5504. 
Specific  Heat.  C#i  =  1.0300  -  0.5490  =  0.4810. 

Example  6.     Water  in  Contact  with  Ice  at  0° 


^  =  0.6400        (||)=-  0.00006       (||)    =  +  0.00011. 
dT  \oTJ  P  \ol  /  p 

Cp2  =  1.0000  water  at  0°. 


80  80 

=  0.6400  -273  +  ^^^t-  0.00006- 

0.00011] 

=  0.6400  -  0.2930  +  0.1510  =  0.4980. 
Specific  Heat.    Cpz  =  1.0000  -  0.4980  =  0.5020. 


THERMODYNAMIC  METEOROLOGY 

Example  7.     Pressure  of  Vapor  in  Contact  with  Water  and  Ice 

P12  =  4.57     vapor-water.  Vi  =  205000. 

p13  =  760      vapor-ice.  v2  =  1.00 

p23  =  water-ice.  v3  =  1.09 

T  =  0.0074°  C.,  the  fundamental  temperature. 


r12         41851000      760 
(295)  Vapor  on  water,  -j^  --  T^_       ~  —  -  1013235- 


606.5    3.1391  X  104 

204999         =  0.3402mm. 


Vapor  on  ice. 


(606.5  +  80)     3.1391  X  104 

273          "       204999     -0.3851mm. 


^0 

(296)       Water  on  ice.    -        - 


80.066      3.1391  X  104 

'  102294'  mm- 


(-0.09) 

Compare  Planck's  "Thermodynamik." 

Application  of  the  Thermodynamic  Formulas  to  the  Non-Adiabatic 

Atmosphere 

The  foregoing  formulas  would  apply  to  an  adiabatic  atmos- 
phere, using  the  constants  of  Table  3,  wherein  Cp,  Cv,  R  are 
constants,  but  they  do  not  apply  to  the  existing  non-adiabatic 
atmosphere,  because  it  is  not  an  ideal  gas,  rather  a  mixture  of 
gases  which  are  undergoing  rapid  changes  of  condition  through 
variations  in  the  heat  contents  by  insolation  and  radiation. 
They  can,  however,  be  adapted  to  the  earth's  atmosphere  by 
suitable  modifications,  which  depend  upon  the  formulas  de- 
veloped under  static  meteorology.  The  following  summary  is 
sufficient  for  working  purposes. 


APPLICATION    TO    NON-ADIABATIC    ATMOSPHERE  81 


Entropy 

T 
jr  -  R 

/QQ1\    e        e        ft  ~  Qo  r>.    TI  —  TQ        1  PI 

(331)  61  -  60  =       rp     ~  =  HI  Cpa  —^  --  TT  Rio  log  -5- 

^ 


dQ  dT          dP  dT       v 

(330)         dS  =   -jr  '•=  Cp-jr  -  R~p~  =  Cp^r  -  j,dP. 


10  JL  10 


M 


Work  Against  External  Forces 

(332)  dW  =  Pdv  =  RdT  -—  =  Cp  ^-^  d  T  -  — . 

p  k  p 

(333)  Wi  -  Wo  =  Pio  (vi  -  VQ)  =  Rw  (Ta  -  T0) L 


Pio 

k  -  1  ,  p, 


Pio 


(334) 


Pio 
Energy 


k-ldP 

=  CpdT  —  —  r  --  . 
k       p 


(335)  £A  -  U0  =  Cvu  (Ta  -  To)  =  (Cpa  -  RIO)  (Ta  -  T0)  = 

Cpa  (Ta  ~  To)  ~  Cpw  ~^  (Ta  -  T0) 
=  (Qi  -  Go)  -  (Wl  -  Wo)   =  Cpa  (Ta  ~  To)  ~ 


pio 
Heat  Energy 

(336)          dQ  =  CvdT  +  RT~    =  Cv  d  T  +  Cp       ^  d  T  - 


k  - 


82  THERMODYNAMIC   METEOROLOGY 

(337)  Q1-Q0  =  Tw  (S,  -So)          =  (Cpa  -  Cpw]  (Ta  -  T0)  = 

Cvw  (Ta  -  T0)  +  Pio  (vi  -  vo). 

=  (Ui  -  Uo)  +  (Wi  -  Wo)  =  Cpa   (Ta  -  To)  - 

Pi-  Po 

Pio 
Radiation  Function 

P-TdS    p-     P*« 
~       ~~         ~f   - 


(Krt   r  >-°          '-»       P  "'-  p 

(339)  *„  =  —  3^  .  =  -^^  -  P10  =        VI_VQ        -  Pm 


(341)  log  K!  -  log  ^o  =  A  (log  r,  -  log  T0). 

(342)  ,  _  log  *i  -  log  go 

(3.42)       -  log  r,  -  log  r.- 

The  Radiation  Coefficients  and  Exponents 

(343)  #10  =  C  TiQA.  log  #10  =  log  C  +  ,4  log  r10. 

(344)  KM  =  c  T10a.  log  #10  =  log  c  +a  log  T10. 

(345)  log  c  =  log  Co  +  (-4  -  4)  log  B. 

C0  =  9.12  X  10  ~5.     B  =  1.66  X  10  ~2. 

(346)  logc  =  -  5.906  -  (2.220)  (A  -  4). 

(347)  c  =CQBA~4  =  9.12  X  10~5(1.66  X  lQ-y~4. 

These  formulas  will  be  fully  explained  and  illustrated  in  the 
examples  that  follow  (pages  84—85). 


Working  Equations 
-Co 


(331)      Si  -  So 


(333)  Wi  -  W0  =  R'w  (Ta  -  To)  - 


Pi  -  Po 

Pio 


WORKING   EQUATIONS  83 

(335)    Ui-U0  =  (Qi-Qo)-(Wi-W0). 
(339)  K10  =  Ul  ~  Uo. 


(340)  =()• 

Ao        Wo/ 


log  Zi  —  log 


In  order  to  illustrate  the  formulas  of  computation  (331) 
to  (342)  the  data  of  Table  17  are  continued  in  Table  19. 
It  must  be  especially  noted  that  -  (Cpa  -  Cpio)  (Ta  -  T0), 
which  by  (198)  includes  the  kinetic  energy  of  circulation  and  the 
kinetic  energy  of  radiation  -  J  (q?  -  q02)  -  (Ql  -  Q0),  is  not 
carried  forward,  but  only  (Qi  —  Q0),  the  energy  of  radiation. 
If  the  former  were  taken  for  the  computations  beyond  this  point 
the  circulation  would  be  treated  as  true  radiation,  which  is 
improper.  The  sign  -  (Ql  -  Q0)  in  Table  17  is  changed  to 
-f  (Qi  —  Qo)  in  Table  19.  In  computing  the  mean  entropy 
from  one  level  to  another,  the  mean  temperature  TiQ  =  f  (T[  -f 
To)  is  taken  from  Table  14  in  successive  pairs.  The  entropy 
generally  increases  with  the  height,  and  always  does  so  unless 
there  is  an  inversion  of  temperature,  or  an  excess  of  wind  varia- 
tion in  velocity  between  the  levels,  such  as  occurred  in  this  case 
between  500  and  1,000  meters. 

In  applying  (333)  for  computing  the  work  (Wi  —  Wo)  we 
must  now  compute  R\o  corresponding  with  (Qi  —  QQ),  which 
differs  from  R1Q  taken  by  pairs  in  Table  14,  since  Rw  implies 
the  circulation  as  well  as  the  radiation.  The  formulas  now  be- 
come 

(348)  (Cpa  -  Cp'lo)  (Ta  -  To)  =  (Qi  -  Qo). 

(349)  Cp\Q  =  Cpa  -  %-^  =  993.58  -  ^=-|°. 

— 


a  —    -  0 


(350) 


84 


THERMODYNAMIC   METEOROLOGY 


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WORKING  EQUATIONS 

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86 


THERMODYNAMIC  METEOROLOGY 


COMPARISON  OF  Rio  WITH  R\ 


Height  z 

116 

500 

1000 

1500 

2000 

2500 

3000 

4000 

5000 

RIO 
R'IO 

287.28 
282.91 

286.87 
287.35 

285.34 
285.65 

283.14 
281.68 

280.50 
277.95 

278.21 
275.46 

275.59 
274.09 

272.06 
268.89 

R'IO  is  generally  smaller  than  Rw  in  these  observations. 

P        P 
The   term  is  taken  directly  from  Table   17,   and 


PlO 


—  Wo  is  easily  computed.     Then  Ui  —  UQ  follows  from  (335). 


In  computing  the  radiation, 


Ui-Ui 


the  values  of 


v  are  the  reciprocal  of  the  density  p  in  Table  14.  Had  p  been 
computed  by  formula  (175),  which  takes  R  constant,  instead  of 
by  (176),  with  R  variable,  it  is  seen  at  once  how  erroneous 
would  have  been  the  derived  radiations,  because  the  values  of 
KIQ  depend  upon  the  small  differences  fyi  —  v0)  in  succession. 
These  radiations  are  mean  values  for  the  strata  concerned.  It 
is  important  to  study  the  relation  of  the  radiation  to  the  tem- 
perature, and  to  compare  the  exponents  of  formula  (340)  with 
the  exponent  of  a  full  radiation  in  the  Stephan  Law,  which  is  4. 
This  subject  is  complex  in  the  earth's  atmosphere  as  will  be  in- 
dicated. The  problem  is  as  follows:  The  values  of  K  in  re- 
lation to  T  by  (340)  are  in  the  form  of  ratios,  whereas  in  the 
Stephan  Law  (344)  they  stand  related  through  a  coefficient. 
If  the  constituent  of  the  ratio  is  in  the  form  (343),  it  is  quite 
certain  that  the  coefficients  (C  .  c}  are  not  equal,  nor  are  the  expo- 
nents (^4  .a).  We  proceed  to  develop  the  relations  between 
C  and  c,  A  and  a.  The  equation  (343)  gives  three  terms,  Kw, 
A,  TIG,  from  which  to  compute  C,  and  it  is  necessary  to  indicate 
what  are  the  relative  values  of  log  C  and  A .  With  the  data  of 
Table  19  in  the  first  section  of  Table  20,  compute  A  log  T10 
and  subtract  this  from  log  KiQ  to  obtain  log  C.  The  negative 
sign  before  the  logarithm  affects  only  the  characteristic.  Thus, 
logarithm  —  11.944  gives  the  number  8.79  X  10~~".  In  this 


WORKING   EQUATIONS 


87 


way  the  values  of  log  C  and  A  were  computed  for  twelve  balloon 
ascensions,  of  which  two  examples  are  given  in  Table  21.  It 
is  readily  seen  that  log  C  is  negative,  as  the  temperature  T  de- 
creases with  the  height  z,  and  positive  in  regions  of  inversion  of 
temperature.  The  magnitude  of  log  C  depends  upon  the  ratio 

rri 

If  the  temperature  changes  slowly  with  the  height  the 

TABLE  20 
COMPUTATION  OF  LOG  C  AND  ( a  .  LOG  c} 

Section  I.     log  C  =  log  Kio  -  A  log  Tio  (343) 
Lindenburg,  April  27,  1909 


z 

TlQ 

log  Tio 

116 

500 
292.25 

1000 

288.30 
2  45984 

1500 

284.90 
2  45469 

2000 

282  .  15 
2  45048 

2500 

279.85 
2  44693 

3000 

277.20 
2  44279 

4000 

272  .40 
2  43521 

5000 

266.00 
2  42488 

log.log  Tio 

0  39091 

0  39000 

0  38925 

0  38862 

0  38789 

0  38654 

0  38469 

log  A 

0.78581 

0  68298 

0  67320 

0  .  68362 

0.67620 

0.55160 

0  62707 

A 

6  11 

4  82 

4  71 

4  83 

4  74 

3  56 

4  24 

log  (A  log 
Tio) 

1.17672* 

1.07298 

1.06245 

1.07224 

1.06409 

0.93814 

1.01176 

A  log  Tio 

15.021 

11.830 

11  547 

11.810 

11  .  590 

8.672 

10  275 

log  Kio 

4  96534 

4  94187 

4  92519 

4  90796 

4  83566 

4  84816 

4  80528 

Kio 

92330 

87472 

84176 

80902 

76853 

70495 

63867 

loeC 

—11  944 

—7  112 

-7  378 

—7  098 

—7  246 

—4  176 

—6  530 

C 

—  ll 
8.79X10 

—  7 

1.29X10 

—  7 

2.39X10 

—  7 

1.25X10 

—  7 

1.76X10 

1.50X10 

3.39X10 

Section  II.     log  c  =  log  Kio  -  a  log  Tio  (344) 


Assumed 

3  82 

3  82 

3  82 

3  82 

3.82 

3.81 

3.81 

log  ai 

0.5821 

0.5821 

0.5821 

0.5821 

0.5821 

0.5809 

0.5809 

log.log  Tio 

0.3909 

0.3900 

0.3893 

0.3886 

0.3879 

0.3865 

0.3847 

log  ai  log 
Tio 

0  9730 

0  .  9721 

0.9714 

0.9707 

0.9700 

0.9674 

0.9656 

9  397 

9  378 

9  363 

9.348 

9.333 

9.277 

9.238 

log  Kio 

4.965 

4  942 

4.925 

4.908 

4.836 

4.848 

4.805 

log  c 

—5  568 

—5  564 

—5  562 

-5.560 

-5.503 

-5.571 

-5.567 

3  824 

3  822 

3  821 

3.820 

3.794 

3.825 

3.823 

Second 
Assumed 

3.807 

3.817 

3.817 

0.5806 

0.5817 

0.5817 

log  log  7\o 

0.3879 

0.3865 

0.3847 

log  .  02  log 

Tio 

0.9685 

0.9682 

0.9664 

9.300 

9.294 

9.256 

log  Kio 

4.836 

4.848 

4.805 

-5.536 

-5.554 

-5.549 

a 

3.809 

3.817 

3.815 

88 


THERMODYNAMIC   METEOROLOGY 

Working  Formulas 


(343)  K1Q  =  C7V 


log  C  =  log  KM  —  A  log  TIG 
(344)  K10  =  c  2V 

log  KIQ  =  log  c  +  a  log  JIG 

log  C   =  log  KIQ  —  a  log  TIQ. 

Assume  trial  #1  and  adjust  by  Table  22.      The  final  pair 
log  c  and  a  should  fall  on  the  same  line. 

TABLE  21       - 

EXAMPLES  OF  LOG  C  .  A  AND  LOG  c  .  a   THE  COEFFICIENTS  AND 
EXPONENTS  IN  THE  RADIATION  FORMULAS  (343),  (344) 


z 

Lindenburg,  April  27,  1909 

Lindenburg,  May  5,  1909 

logC 

A 

logc 

a 

logC 

A 

logc 

a 

17000 

16000 

+28  .  034 

—  10  .  34 

—  5.435 

3.764 

15000 

+52.871 

-21.11 

-5.461 

3.775 

14000 

+  16.828 

—  5.78 

-5.489 

3.788 

13000 

-  4.356 

3.45 

-5.531 

3.807 

+  10.437 

-  3.41 

-5.537 

3.810 

12000 

-  7.140 

4.86 

-5.563 

3.822 

-27.533 

13.29 

-5.563 

3.821 

11000 

-  5.962 

3.64 

-5.571 

3.825 

-  6.181 

4.40 

-5.548 

3.814 

10000 

-   1.410 

3.44 

-5.560 

3.820 

-  3.288 

3.09 

-5.558 

3.819 

9000 

-  2.114 

2.73 

-5.549 

3.815 

-  4.021 

3.62 

-5.558 

3.819 

8000 

-  3.211 

3.11 

-5.539 

3.810 

-  6.506 

4.26 

-5.560 

3.820 

7000 

-  7.178 

4.79 

-5.538 

3.810 

-  5.249 

3.95 

-5.557 

3.818 

6000 

-  9.719 

5.40 

-5.548 

3.814 

-  5.401 

3.89 

-5.560 

3.820 

5000 

-  6.530 

4.24 

-5.549 

3.815 

-11.132 

6.48 

-5.547 

3.814 

4000 

-  4.176 

3.56 

-5.554 

3.817 

-  4.050 

3.63 

-5.560 

3.820 

3000 

-  7.246 

4.74 

-5.536 

3.809 

-  8.115 

5.25 

-5.567 

3.823 

2500 

-  7.098 

4.83 

-5.560 

3.820 

-18.387 

9.23 

-5.577 

3.828 

2000 

-  7.378 

4.71 

-5.562 

3.821 

-20.982 

9.81 

-5.590 

3.833 

1500 

-  7.112 

4.82 

O  .  OvVx 

3.822 

-14.180 

7.64 

-5.601 

3.838 

1000 

-11.944 

6.11 

-5.568 

3.824 

+24,708 

-8.29 

-5.593 

3.834 

500 

ratio  is  small.     At  the  same  time  the  ratio   h~  J ,  which  does 

\Ar,/ 

not  have  a  coefficient  or   exponent,  registers  a  change  in  the 


WORKING  FORMULAS 


89 


atmosphere  which  is  closely  connected  with  the  variation  of 
the  pressure  P.  Hence  there  are  large  changes  in  log  C  and  A 
which  are  opposite  in  sign,  but  both  increasing  or  diminishing 
together.  Under  nearly  normal  conditions  it  is  seen  that  A  is 
approximately  4.00,  which  would  be  the  value  for  a  full  radiating 
black  body.  The  entire  series  of  values  A,  log  C,  were  collected 
in  groups,  and  the  mean  values  when  plotted  fall  on  a  straight 
line,  of  which  the  equation  was  found  to  be, 

(347)        C  =  C0BA~\ 

(345)  log  C  =  log  Co  +  (A  -  4)  log  B. 

(346)  logC  =  -  5.960  +  (A  -  4)  (-  2.220). 

(347)  C  =  9.12  X  10~5  (1.66  X  10  ~2)A^4. 

The  development  of  a  portion  of  this  formula  from  A  =  4.00 
to  A  =  3.50  is  given  in  Table  22.  A  =  4.00  corresponds 
with  a  full  black  radiator  and  A  =  3.50  corresponds  with  the 
theoretical  value  for  the  atmospheric  air.  The  mean  value  was 
found  to  be  A  =  3.82  near  the  surface.  The  notation  (log  c  . 
a)  indicates  the  values  in  the  constituent  formula,  while  (log  C  . 
A)  are  used  for  the  ratio  formula. 

TABLE  22 
EVALUATION  OF  THE  FORMULA  (346) 


a     logc 

a      log  c 

a     log  c 

a      logc 

4.00  -5.960 
3.99  -5.938 
3.98  -5.916 
3.97  -5.893 
3.96  -5.871 

3.95  -5.849 
3  94  —5  827 

3.85  -5.627 
3.84  -5.605 
3.83  -5.583 
3.82  -5.560 
3.81  -5.538 

3.80  -5.516 
3  79  _5  494 

3.70  -5.294 
3.69  -5.272 
3.68  -5.250 
3.67  -5.227 
3.66  -5.205 

3.65  -5.183 
3  64  -5  161 

3.55  -6.961 
3.54  -6.939 
3.53  -6.917 
3.52  -6.894 
3.51  -6.872 

3.50  -6.850 

3.93  —5.805 

3.78  -5.472 

3.63  -5.139 

3.92  -5.782 

3.77  -5  449 

3.62  —5.116 

3  91  —5  760 

3  76  —5  427 

3  61  —5  094 

3.90  -5.738 

3  75  —5  405 

3  60  -  5  072 

3.89  -5.716 
3.88  -5  694 

3.74  -5.383 
3  73  —5  361 

3.59  -5.050 
3  58  -5  028 



3.87  -5.671 

3.72  -5.338 

3.57  —5.005 

3.86  -5  649 

3  71  -5  316 

3  56  -6  983 

90  THERMODYNAMIC   METEOROLOGY 

The  negative  sign  applies  only  to  the  characteristic 
log  Co  =  -  5.960     Co  =  9.12  X  10~5 
logB  =  -  2.220    B  =  1.66  X  10~2. 

By  means  of  Table  22  we  proceed  in  Section  II  of  Table 
20  to  compute  log  c  and  a  from  log  C  and  A .  Assume  an 
approximate  value  ai,  as  3.82,  and  compute  di  log  7\0  from  the 
value  in  Section  I.  Subtract  from  log  K\Q  for  log  c,  and  in 
Table  22  interpolate  that  value  a0,  which  is  the  pair  value  of 
log  c.  If  this  value  aQ  agrees  with  the  assumed  a\  the  check  is 
complete.  If,  on  the  other  hand,  these  values  of  logc  and  a 
do  not  quite  agree,  as  in  the  examples  under  z  =  3000,  4000,  5000, 
take  the  mean  value  between  the  assumed  a\  and  computed 
aQ,  and  proceed  again  with  a2  to  compute  logc  and  a.  The 
second  trial  is  usually  successful  if  a\  has  been  chosen  with  some 
practice.  The  corresponding  values  of  (log  c  .  a)  are  found  in 
the  examples  of  Table  21,  and  it  is  seen  that  the  irregularities 
of  (log  C  .  A)  have  disappeared.  Log  c  and  a  usually  decrease 
slowly  with  the  elevation  and  with  the  increase  of  latitude  from 
the  equator. 

These  results  check  by  log  c  +  a  log  TIQ  =  log  KIQ. 

Application  of  the  Thermodynamic  Formulas  to  Various  Meteor- 
ological Problems 

It  has  seemed  necessary  to  give  an  extended  example  of  the 
method  of  computing  the  thermodynamic  values  in  the  non- 
adiabatic  atmosphere,  on  account  of  the  complexity  of  the 
computations,  and  because  of  the  numerous  valuable  results 
dependent  upon  them.  In  Bulletin  No.  3  of  the  Argentine 
Meteorological  Office,  1913,  will  be  found  the  results  for  many 
types  of  data  in  considerable  detail.  We  can  here  summarize 
them  briefly,  depending  upon  diagrams  to  bring  out  the  general 
ideas,  in  particular  respecting  the  isothermal  region,  the  diurnal 
convection,  the  circulation  in  cyclones  and  anti-cyclones,  and 
the  general  circulation  of  the  atmosphere. 


THE    ISOTHERMAL    REGION  91 

The  Isothermal  Region 

It  has  been  found  by  balloon  ascensions  to  great  elevations, 
up  to  20,000  meters  or  more,  that  the  temperature  of  the  atmo- 
sphere diminishes  at  the  rate  of  about  6.0°  C.  per  1,000  meters 
up  to  an  elevation  of  12,000  meters  in  Europe,  or  15,000  meters 
in  the  tropics,  or  even  to  20,000  meters  over  the  equator,  while 
above  these  elevations  the  temperature  is  nearly  constant  or 
increases  a  little  to  the  highest  levels  explored.  There  have 
been  many  conjectures  as  to  the  cause  of  the  permanence  of  the 
heat  of  this  isothermal  region,  as  overflow  of  the  tropic  heat 
to  mid-latitudes,  conductional  transportation  of  heat  from  the 
lower  to  the  higher  levels,  production  of  ozone  by  the  incoming 
solar  radiation  in  the  upper  atmosphere  and  absorption  of  the 
short  waves  of  the  solar  radiation  in  the  same  region.  There  are 
objections  to  each  of  these  hypotheses  so  obvious  that  we  pro- 
ceed at  once  to  examine  the  thermodynamic  data  for  at  least  a 
statement  of  the  case,  if  not  a  complete  explanation  of  the  facts. 

The  computations  were  executed  for  the  following  balloon 
ascensions,  as  reported  in  the  volumes: 

Europe  Atlantic  Tropics 

Lindenburg,  April  27, 1909  (52°).  Sept.  25, 1907  (35°). 

May  5,  1909     "  Sept.  9, 1907  (25°). 

May  6,  1909     "  Aug.  29, 1907  (13°). 

July  27, 1908     "  July  29, 1907  (13°). 

Sept.  2, 1909     "  June  19, 1906  (-2°). 

Mailand,       Sept.  7, 1906  (45°).  Victoria  Nyanza,  1908  (0°). 

The  mean  values  are  compiled  in  Table  23,  and  illus- 
trated in  Figs.  5  and  6.  Since  the  data  in  the  isothermal  region 
are  not  so  complete  as  below  it,  these  results  are  to  be  considered 
as  instructive  rather  than  definitive.  It  will  require  the  work 
of  many  years  to  accumulate  and  compute  the  data  necessary 
for  normal  conditions.  The  temperatures  show  that  there 
are  as  wide  local  fluctuations  in  the  isothermal  region  as  below 
it.  Furthermore,  the  temperatures  are  lowest  over  the  equator, 
200°,  and  gradually  increase  to  210°  in  the  tropics,  or  215°  in 


92 


THERMODYNAMIC   METEOROLOGY 


Europe.  From  these  minimum  values  the  temperatures  in 
the  isothermal  region  increase  about  10  degrees.  These  facts 
appear  clearly  in  the  Table  23,  and  the  diagrams,  where 


13000 


10000 


5000 


000 


P-  K 


Pressure 

Radiation 

Energy 


Density 


25000 


50000 


75000 


100000 


0.7500 


1.0000 


1.2500 


1-5000 


10000 


5000 


000 


200C 


Temperatu  e 


\ 


Free  Heat 


250°  275°  -3000 

Full  .Line  for  Europe 
DottedJLine  for  Tropics 
FIG.  5 


-2000 


-1000 


the  other  data  are  presented.  The  data  on  Fig.  5  (P,  —K,  p, 
Qi  —  QQ)  may  be  considered  primary,  and  those  on  Fig.  6 
(Cp,  Si  -  So,  Wi  -  WQ,  Ui  -  UQ)  secondary,  as  being  the 
machinery  of  the  thermal  engine.  It  is  noted  that  (P,—K,  p) 


THE    ISOTHERMAL  REGION 


93 


have  one  configuration,  and  (Q,  Cp,  S,  W,  U)  another  configura- 
tion. The  former  is  more  immediately  under  the  control  of 
gravitation  acting  downward,  and  the  latter  is  the  result  of 


15000 


5000 


000 


Cp       Specific  Heat 


En 


tropy 


\ 


600  700 


800  900  1000  -10.00          -5.00 


15000 


10000 


5000 


Work 


-  U0      Inner  Energy 


000 
4000  5000  6000  7000  -8000  -7500  -7000          -6500 

Full  Line  for  Europe 
Dotted  Line  for  Tropics 
FIG.  6 

radiation.    The  circulation  q  is  the  balancing  governor  to  the 
engine  which  keeps  the  other  two  parts  in  equilibrium. 

(<2i  -  Qo).  There  is  a  persistent  supply  of  heat  from  four 
conditions:  (l)  That  from  the  heated  earth  as  the  source; 
(2)  that  in  the  lower  strata,  due  to  convection  within  2,000 


94 


THERMODYNAMIC  METEOROLOGY 


TABLE  23 

MEAN  THERMODYNAMIC  VALUES  FOR  EUROPE 


z 

T 

P 

p 

Cp 

g 

Qi-Qo 

Si  -So 

Wi- 

W0 

Ui-U 

KlQ 

20000 
19000 
18000 

17000 

219.5 

9141 

0.2220 

657.15 

0.8 

-3231 

-14.747 

4658 

-7887 

16704 

16000 

223.4 

10983 

.2476 

689.35 

2.5 

-2791 

-12.620 

4962 

-7753 

18497 

15000 

221.2 

12819 

.2764 

731.44 

3.9 

-2392 

-11.158 

5209 

-7602 

20083 

14000 

220.3 

14972 

.3086 

769.99 

8.6 

-1955 

-  8.641 

5540 

-7494 

21850 

13000 

215.7 

17017 

.3422 

799.60 

14.3 

-1702 

-  7.487 

5730 

-743l!23883 

12000 

215.8 

19942 

.3830 

836.65 

16.4 

-1381 

-  6.326 

5948 

-7329J26548 

11000 

218.7 

23341 

.4283 

863.20 

17.8 

-1190 

-  5.345 

6096 

-7285 

29953 

10000 

226.0 

27198 

.4780 

873.16 

19.3 

-1089 

-  4.722 

6164 

-7253 

34502 

9000 

232.8 

31606 

.5315 

884.82 

20.7 

-1131 

-  4.768 

6248 

-737940630 

8000 

241.1 

36509 

.5890 

890.49 

18.4 

-1034 

-  4.229 

6302 

-733645275 

7000 

249.0 

41972 

.6504 

897.75 

16.3 

-  920 

-  3.647 

6329 

-7249 

51345 

6000 

256.1 

48053 

.7162 

907.35 

12.6 

-  793 

-  3.083 

6405 

-7198 

57729 

5000 

262.6 

54819 

.7865 

919.62 

14.1 

-  650 

-  2.578 

6511 

-7160 

64352 

4000 

268.3 

62353 

.8620 

934.08 

13.0 

-  530 

-  1.957 

6601 

-7131 

71739 

3000 

274.3 

70722 

.9429 

947.41 

12.6 

-  380 

-  1.375 

6717 

-7097 

77430 

2000 

279.1 

80030 

1.0299 

964.88 

11.8 

-  241 

-  0.857 

6825 

-7066 

86011 

1000 

284.8 

90345 

1.1224 

979.50 

9.2 

-  76 

-  0.241 

6969 

-7045 

94582 

100 

289.4 

100419 

1.2101 

993.58 

6.1 

meters  of  the  ground,  and  involving  a  supply  of  latent  heat  by 
the  condensation  of  aqueous  vapor  into  water  in  cloud  formation 
by  (295);  (3)  that  in  the  cirrus  cloud  region,  9,000  to  15,000 
meters  elevation,  according  to  the  latitude,  due  to  ice  formation 
from  frozen  water  or  vapor  by  (296).  In  this  cirrus  region  there 
are  other  sources  of  heat  supply,  such  as  an  accumulation  of 
heat  from  absorption  of  radiation  producing  the  new  rate  of 
loss  of  free  heat  per  1,000  meters.  Take  the  differences  in  the 
((?i  —  (?o)  columns,  and  the  mean  values  fall  into  two  groups, 
omitting  those  in  the  cirrus  layers. 

Europe,  12000  to  17000  A  (Ql  -  Q0)  =  -  370  per  1000  meters. 

Tropics,  14000  to  17000  =  -  476 

Europe,  1000  to  11000  =-133 

Tropics,  1000  to  13000  "  =  -  140        "  " 


THE   ISOTHERMAL   REGION 


95 


TABLE  23 
MEAN  THERMODYNAMIC  VALUES  FOR  THE  ATLANTIC  TROPICS 


a 

20000 
19000 
18000 

r 

P 

p 

Cp 

<? 

Qi-Qo 

Si  -So 

Wi- 
Wo 

Ui-Ut 

Kio 

17000 

227.0 

9894 

0.2256 

668.93 

8.0 

-3012 

-13.359 

4911 

-7922 

17488 

16000 

217.3 

11213 

.2513 

709.00 

7.4 

-2570 

-10.292 

5203 

-7774 

18647 

15000 

209.9 

13331 

.2788 

788.87 

8.8 

-1898 

-  8.567 

5609 

-7506 

19328 

14000 

212.8 

15665 

.3128 

814.91 

8.5 

-1584 

-  7.361 

5750 

-7334 

21544 

13000 

215.9 

18398 

.3502 

835.84 

14.3 

-1470 

-  6.697 

5911 

-7381 

24731 

12000 

221.6 

21442 

.3911 

855.58 

13.8 

-1345 

-  5.949 

6024 

-7369 

28286 

11000 

230.3 

24942 

.4355 

861.08 

13.3 

-1302 

-  5.545 

6060 

-7362 

32588 

10000 

238.9 

28853 

.4830 

865.62 

12.4 

-1250 

-  5.135 

6086 

-7336 

37288 

9000 

247.7 

33204 

.5338 

869.26 

12.0 

-1182 

-  4.699 

6127 

-7309 

42938 

8000 

254.9 

38038 

.5879 

878.51 

11.8 

-1100 

-  4.252 

6199 

-7298 

47822 

7000 

260.1 

43413 

.6459 

887.63 

10.7 

-1000 

-  3.769 

6271 

-7271 

53935 

6000 

268.3 

49379 

.7079 

899.98 

9.9 

-  881 

-  3.247 

6368 

-7248 

59582 

5000 

274.1 

56008 

.7742 

913.55 

6.6 

-  700 

-  2.534 

6473 

-7173 

65980 

4000 

278.5 

63380 

.8453 

931.88 

6.5 

-  532 

-  1.907 

6591 

-7123 

72304 

3000 

283.9 

71569 

.9217 

946.73 

6.5 

-  397 

-  1.388 

6694 

-7091 

78606 

2000 

288.8 

80625 

1.0031 

963.44 

5.8 

-  187 

-  0.641 

6846 

-7033 

86055 

1000 

292.4 

90699 

1.0908 

984.35 

4.4 

-  24 

-  0.081 

6911 

-6935 

93656 

000 

299.5 

101753 

1.1837 

993.58 

5.8 

In  the  Europe  group  9,000  to  10,000  is  omitted,  and  in  the 
tropic  group  10,000  to  12,000  is  omitted,  as  being  regions  of 
special  local  supply.  It  appears  that  heat  is  lost  at  about  three 
times  greater  rate  in  the  isothermal  region  than  in  the  lower 
levels.  This  occurs  at  the  same  time  the  temperature  is  rising 
in  the  isothermal  region  but  falling  in  the  lower  levels.  The 
Victoria  Nyanza  ascensions  give  A  (Qi  —  Q0)  =  —  144  through- 
out the  region  3,000  to  18,000,  but  in  this  case  no  inversion 
of  T  was  found.  (4)  The  principal  fact  to  be  explained  is  the 
slow  rate  of  loss  of  heat  in  the  convectional  region,  140  per 
1,000  meters,  as  compared  with  that  in  the  isothermal  region, 
about  400.  This  is  easily  accounted  for  by  the  following  facts : 
The  incoming  solar  radiation  is  of  short  wave  lengths,  and  pen- 
etrates to  the  earth's  surface,  having  only  a  small  amount  of 


96  THERMODYNAMIC   METEOROLOGY 

selective  absorption  of  radiant  energy.  This  is  transformed  at 
the  earth  into  long  waves,  in  changing  the  temperature  energy 
from  7,000°  to  300°,  and  this  heat  escapes  to  space  partly  by  radi- 
ation and  partly  by  vertical  convection,  the  latter  extending  to 
the  isothermal  layer,  whose  height  varies  with  the  latitude,  and 
the  heat  contents  of  the  air  at  the  surface.  In  the  general  ver- 
tical convection  of  the  atmosphere,  as  the  temperature  of  a 
unit  mass  changes  from  T0  to  7\,  in  the  vertical  distance  z\  —  z0, 
there  is  an  evolution  of  heat  Qi  —  Qo  =  Cp1Q  (T\  —  jT0),  which 
is  added  to  the  atmosphere  throughout  the  convectional  region. 
If  (Qi  —  Qo)  is  the  natural  loss  of  heat  by  radiation  without 
convection,  and  ((V  —  QQ}  the  amount  evolved  by  convective 
cooling  of  the  temperature  of  the  rising  mass,  then  we  have, 
400  =  (Qi  —  Qo)  the  heat  loss  in  the  isothermal  region,  and  140  = 
(Qi  ~~  Qo)  ~~  (Qi  ~  Qo)  that  in  the  convection  region.  Hence 
260  =  (Qi  —  Qo)  the  heat  evolved  by  vertical  motion.  This 
subject  will  require  a  fuller  development  than  is  at  present 
available,  and  it  is  complicated  by  the  fact  that  the  vertical 
distance  through  which  the  mass  moves  is  not  well  known.  The 
air  that  has  risen  by  convection  with  cooling  and  evolution  of 
heat  in  one  place  falls  again  in  other  places  with  heating  and  ab- 
sorption of  heat.  Such  places  of  descending  air  are,  during  the 
night,  in  the  permanent  high-pressure  belts  and  in  the  wander- 
ing anti-cyclones.  This  subject  will  be  sufficiently  illustrated 
in  the  following  sections. 

P,  -  K  and  p. 
Pressure,  Radiation-Potential,  and  Density. 

The  pressure  is  found  to  change  continuously  in  a  smooth 
curve  from  the  surface  upward,  that  in  the  tropics  being  some- 
what higher  in  value  than  in  Europe.  An  entirely  similar 
curve  is  developed  by  the  potential  radiation  —  K,  the  values 
being  always  higher  than  P.  The  density  is  also  given  in  a 
smooth  curve,  the  tropics  and  Europe  crossing  at  about  7,000 
meters.  It  would  seem,  then,  that  the  ultimate  purpose  of 
T.  Q  and  the  other  dependent  terms  is  to  so  regulate  the  pressure 


PRESSURE,   RADIATION-POTENTIAL,    AND   DENSITY  97 

and  radiation  that  they  shall  change  steadily  from  level  to  level, 
under  the  attraction  of  the  earth's  gravitation,  and  that  all  the 
other  thermodynamic  values  mutually  adjust  themselves  to 
produce  this  simple  result.  Hence,  the  problem  depends  upon 
the  rate  of  loss  of  A  (Qi  —  Q0)  in  the  lower  and  the  isothermal 
regions,  which  is  distinctly  a  physical  question.  A  reason  has 
been  already  indicated  why  the  rate  of  transfer  of  heat  should 
be  greater  in  one  region  than  in  the  other,  and  why  the  iso- 
thermal separates  from  the  convectional  region. 
By  (339)  we  have  the  following  equation: 


A«<n  K    . 

(339)  K10  - 


This  equation  is  immediately  derived  from   (259),   where 

C\  /^ 

CT  =  (  —  J    is  the  latent  heat  while  the  temperature  is  constant; 

it  also  comes  from  (309),  where  the  latent  heat  r2  of  vaporization 
is  derived  by  primary  analysis;  or  it  may  be  taken  from  (338), 
(339);  finally  it  is  found  by  computation  that  the  last  form 
through  the  gas  coefficients  Ra,  RIO  is  satisfied,  as  can  easily  be 
verified  by  the  data  of  Tables  14  and  19.  The  small  dis- 
crepancies are  due  entirely  to  the  velocity  term  which  was 
eliminated  by  means  of  Table  17.  —  KIQ  is  negative  in  sign 
because  (U\  —  U0)  is  negative  while  fyi  —  v0)  is  positive.  It 
seems,  then,  that  the  latent  heat, 


Hence,  the  escape  of  heat  or  radiation  in  the  earth's  atmosphere 
depends  entirely  upon  the  divergence  of  the  gas  coefficient  ^10 
from  the  adiabatic  value  Ra,  as  was  stated.  The  divergence  of 
the  lines  P10  and  —  K^  in  Fig.  5  measures  this  term.  If  the 
velocity  is  also  considered  it  will  be  a  line  near  the  —  KIQ  line, 
slightly  adjusting  it  to  make  the  pressure  transitions  gradual. 
This  is  the  function  of  the  horizontal  cloud  motions  of  flowing 


98  THERMODYNAMIC  METEOROLOGY 

strata  so  generally  seen  in  the  atmosphere.  This  confirms  the 
principle  of  equations  (36)  to  (38),  which  indicate  the  relations 
of  pressure,  circulation,  and  radiation  to  gravitation. 

It  is  easy  to  see  that  such  data  are  capable  of  making  all  the 
general  thermodynamic  formulas  (205)  to  (328),  and  many  others, 
applicable  to  the  earth's  non-adiabatic  atmosphere.  It  should 
be  carefully  noted  that  the  density  p,  and  the  gas  coefficient  R, 
must  be  computed  by  (176),  (177),  and  not  by  (175)  for  Ra  con- 
stant; that  the  effective  specific  heat  Cp  is  variable,  and  that 
radiation  depends  upon  this  fact.  The  principal  quantities  to 
obtain  by  observation  are  the  temperature  T,  and  the  velocity 
of  circulation  q  at  the  height  z,  and  hence  the  observations  for 
temperature  alone,  omitting  q,  are  not  capable  of  giving  correct 
radiation  data.  Finally,  the  variation  of  pressure  —  d  P  is  not 
proportional  to  the  mass  gpdz  =  gdm,  but  by  (201)  the  terms 
p  q  d  q  +  p  d  Q  must  be  added  for  circulation  and  radiation,  or 
else  P  =  —  K,  which  is  to  exclude  them  from  the  problem,  and 
reduce  it  to  the  unusual  adiabatic  case.  One  can  now  perceive 
that  there  is  no  possibility  of  solving  the  general  equations  of 
motion  in  cyclones  and  anti-cyclones,  and  in  all  the  other  types 
of  circulation,  without  first  eliminating  the  heat  term  d  Q.  Nearly 
all  attempts  of  meteorologists  to  solve  the  circulation  problems 
have  been  futile  chiefly  on  this  account,  because  of  the  assumed 
necessity  of  ascribing  to  friction,  and  to  the  deflecting  force  of 
the  earth's  rotation  on  a  moving  mass,  values  which  they  do  not 
actually  possess.  We  shall  be  able  to  explain  this  more  fully 
in  the  chapter  on  Dynamic  Meteorology,  but  now  proceed  to 
illustrate  more  at  length  the  thermodynamic  terms  in  other 
typical  conditions  of  the  atmosphere. 

The  Diurnal  Convection  and  the  Semi-diurnal  Waves  in  the  Lower 

Strata 

There  is  a  series  of  problems  relating  to  the  semi-diurnal 
waves  observed  at  the  surface,  which  have  been  much  discussed 
without  satisfactory  results,  as  the  semi-diurnal  barometric 
waves  and  the  several  electrical  and  magnetic  waves  which  are 


CONVECTION  AND   SEMI-DIURNAL  WAVES  99 

associated  with  the  diurnal  convection.  At  the  surface  the 
temperature  has  only  a  single  diurnal  wave,  and  it  has  not  been 
possible  to  match  these  two  series  of  data  in  a  definite  relation 
of  cause  and  effect.  The  difficulty  in  studying  the  general 
problem  has  been  the  lack  of  observations  in  the  free  air  above 
the  surface,  especially  during  the  night.  The  only  exception 
to  this  defect  is  the  series  of  kite  ascensions  at  the  Blue  Hill 
Observatory,  1897-1902,  which  were  discussed  in  my  papers, 
Monthly  Weather  Review,  February  to  August,  1905.  In  these  it 
was  shown  that  the  single  diurnal  temperature  wave  changes 
into  a  semi-diurnal  wave  at  about  400  meters  above  the  surface, 
and  that  the  semi-diurnal  waves  die  away  within  two  or  three 
thousand  meters  of  the  ground.  It  was  also  indicated  that  the 
other  data,  namely,  vapor  pressure,  atmospheric  electric  potential, 
ionization,  and  magnetic  fields,  have  diurnal  variations  closely 
matching  the  diurnal  circulation.  This  section  gives  the  result 
of  another  discussion  of  the  subject,  using  the  data  of  the  Cordoba 
and  Pilar  stations,  Argentina. 

There  are  two  hypotheses  regarding  the  origin  of  the  semi- 
diurnal barometric  waves:  (1)  The  forced  oscillation  of  the 
entire  atmosphere,  as  proposed  by  Lord  Kelvin,  and  developed 
by  Margules,  Hann,  Jaerisch,  Gold,  and  others;  (2)  the  effect 
of  the  diurnal  convection,  proposed  in  general  terms  by  Espy, 
and  studied  by  Ferrel,  Koppen,  Sprung,  Bigelow,  and  others. 
The  important  objections  to  the  former  hypothesis  are,  that 
these  waves  do  not  embrace  the  entire  atmosphere  and  are 
usually  confined  within  2,000  meters  of  the  surface;  that  the 
analytical  equations  and  harmonic  analyses  merely  represent, 
in  other  forms,  the  data  assumed  for  their  coefficients,  and  do 
not  reach  the  origin  of  the  physical  causes;  and  that  these 
equations  have  not  the  radiation  and  heat  terms,  which  are 
more  important  than  the  friction  and  the  deflecting  force  of 
rotation. 

The  method  of  treating  the  Cordoba-Pilar  data  was  to  assume 
a  uniform  temperature  T,  and  pressure  P,  on  the  3,500-meter 
level,  and  then,  by  studying  the  observed  temperatures  and  wind 
velocities  on  several  levels,  000,  200,  400  ...  3,500  meters,  at 


100 


THERMODYNAMIC  METEOROLOGY 


Scale 
AT 


0  p.m. 


2a.m. 


6a.m. 


10  a.m. 


2p.m. 


6p.m. 


10  p.m. 


2a.m. 


2500 


2000 


1500 


1000 


400 
200 


000 


+  3.0 

+  2.0 

+  1.0 

0 

-1.0 
-2.0 
-3.0 
-40 


280.9 


284.7 


NN. 


291.5 
293.3 
295.0 
296.3 
296.4 
295.8 


FIG.  7.     A  T — Temperature  of  the  semi-diurnal  waves  above  400  meters  and 

the  diurnal  wave  near  the  surface 
T  is  the  mean  temperature  on  the  given  level. 


AP 


.10  p.mj    2  a.m. 


6a.m. 


10  a.m. 


2p.m. 


6p.m. 


10  p.m. 


2a.m. 


2500 


2000 


1500 


1000 
800 
600 
400 
200 
000 


75463 


Units 
+160 
-1-120 
f  80 
+  40 
0 

—  40 
-80 
-120 
-160 


\X 


85087 


90253 
92387 
94555 
96771 
99027 
101338 


FIG.  8.    A  P — Pressure  of  the  semi-diurnal  waves  in  all  these  strata  vanish- 
ing at  about  the  level  2500  meters 


CONVECTION  AND   SEMI-DYURNAi;  ^WAVES' J 


101 


the  hours  (2,  6,  10)  A.M.,  (2,  6,  10)  P.M.,  proceed  by  computations 
entirely  similar  to  those  of  Tables  14,  17,  19,  to  derive  a 
pressure  P0  at  the  surface  which  would  exactly  match  the  mean 


AB 

10p.m.         2a.m.            6a.m.          10a.m.          2p.m.          6p.m.           10p.m.           2a.m. 

+  1.00 
0.00 
-1.00 

_~n 

^^ 

^ 

\ 

fr 

'^ 

\ 

\ 

/ 

w»n  T. 

ne=Ot»se 
Line  =  Cc 

rved  Data 
mputed  D 

^VJ 

ita 

/ 

Dotted 

FIG.  9.     Computed  and  observed  pressure  waves  at  the  surface 


10  p.m. 


2  a.m. 


6  a.m. 


10  a.m. 


2  p.m. 


6  p.m. 


10  p.m. 


2a.m.  AQ 


-70 


-70 


2500 


2000 


1500 


-50 


1000 


800 


-50 


-40 


600 


400 
200 
000 


-30 

-20 

-10 

0 

+  10 
+  20 


y/ 

0+20 


FIG.   10.    The  loss  of  heat  for  every  200  meters  A  Q  =  (Ql  —  Q0) 

pressure  as  observed  at  Cordoba.  It  required  several  trial 
computations  to  accomplish  this  result,  especially  during  the 
night  hours  where  direct  observations  were  lacking  above  the 


102  TIIERVOD  ^NAMIC  METEOROLOGY 

surface,  and  the  outcome  is  indicated  in  Tables  24  and  25, 
and  in  Figs.  7,  8,  9,  10.  All  the  other  data,  as  computed,  will 
be  found  in  Bulletin  No.  3,  O.  M.  A.  Transferring  the  data  of 
Table  24  to  the  diagrams,  it  is  seen  that  the  single  diurnal 
temperature  wave  at  the  surface  transforms  into  semi-diurnal 
waves  at  400  meters,  and  that  these  die  away  at  2,500  meters. 
This  result  confirms  the  discussion  of  the  Blue  Hill  data  in  all 
respects.  The  corresponding  pressure  waves  are  semi-diurnal 
throughout  these  strata  from  the  surface  to  2,500  meters,  and 
there  they  vanish.  The  night  wave  is  weaker  than  the  day  wave 
in  consequence  of  the  temperature  inversion  near  the  surface, 
both  waves  being  nearly  equal  at  1,000  meters. 

Since  the  barometric  variations  are  alone  of  interest,  the 
base  line  has  been  taken  as  that  near  sea  level.  The  discrepancy 
at  2  P.M.  is  due  to  an  imperfect  temperature  distribution  at  that 
hour,  which  should  be  made  cooler  by  a  few  tenths  of  a  degree. 

By  adjusting  the  vertical  temperatures  a  little  it  would  be 
possible  to  reproduce  the  observed  curve  with  precision  through 
the  non-adiabatic  computations.  There  are  no  other  tempera- 
tures which  could  reproduce  this  pressure  wave  at  the  surface, 
and  this  fact  is  proof  of  the  cause  of  the  observed  pressure  system. 
In  order  to  understand  more  fully  the  origin  of  the  temperature 
system,  the  values  of  A  (Qi  —  QQ),  the  variations  on  the  daily 
mean  are  plotted  on  Fig.  10  and  the  curves  of  equal  heat  losses 
are  drawn.  The  data  are  somewhat  imperfect,  but  the  general 
result  is  not  doubtful.  It  shows  that  there  are  two  principal  axes 
of  heat  exchange,  that  in  the  afternoon,  2  P.M.,  at  the  surface,  to 
8  P.M.,  at  2,500  meters,  and  that  in  the  night  from  1  A.M  at  2,500, 
to  4  A.M.,  at  the  surface.  That  is  to  say,  the  air  rises  obliquely  in 
the  afternoon  to  the  right  and  falls  in  the  night,  also  to  the  right. 
The  air  rises  and  falls  in  such  a  zigzag  path  as  gives  a  turning- 
point  at  about  10  P.M.  above  and  10  A.M.  below,  judging  by  the 
crests.  The  rising  air  cools  by  expansion,  and  the  falling  air 
heats  by  compression,  the  former  producing  the  afternoon 
wave  and  the  latter  the  night  wave  to  within  400  meters  of  the 
surface  on  Fig.  7.  At  this  level  the  more  rapid  cooling  of  the 
ground  during  the  night  makes  itself  felt,  and  there  is  radiation 


CONVECTION   AND    SEMI-DIURNAL   WAVES  103 

TABLE  24 

THE  SEMI-DIURNAL  TEMPERATURE  WAVES  IN  THE  STRATA  400  TO  2,500 
METERS,  TOGETHER  WITH  THE  SINGLE  DIURNAL  TEMPERATURE  WAVE 
AT  THE  SURFACE 


z 

r 

P 

2 

-02H2o) 

T 

P 

2 

-02H2o) 

2  A. 

VI. 

6  A 

.M. 

2500 

281°.  0 

75464 

3.0 

+  151.8 

280°.  8 

75466 

6.0 

+  140.2 

2000 

285.  0 

80153 

4.0 

+131.9 

284  .2 

80162 

7.0 

+120.1 

1500 

288  .6 

85080 

5.0 

+  111.9 

287  .5 

85104 

7.0 

+  97.6 

1000 

291  .9 

90230 

5.0 

+  38.5 

290  .6 

90288 

6.0 

+  34.9 

800 

293  .6 

92368 

5.0 

+  40.2 

292  .3 

92428 

5.0 

+  33.1 

600 

295  .0 

94530 

4.0 

+  30.0 

294  .3 

94604 

4.0 

+  25.8 

400 

295  .8 

96743 

4.0 

+  16.1 

295  .4 

96820 

4.0 

+  13.2 

200 

293  .7 

99018 

3.0 

-  11.7 

293  .4 

99095 

3.0 

-  15.8 

000 

231  .7 

101349 

2.0 

291  .0 

101440 

2.0 

10  A. 

M. 

2p 

M. 

2500 

280°.  7 

75462 

9.4 

+  144.2 

280°.  9 

75462 

9.7 

+  146.7 

2000 

284  .0 

80173 

9.4 

+  114.2 

284  .8 

80162 

10.3 

+133.0 

1500 

287  .0 

85112 

9.4 

+  87.4 

288  .8 

85084 

10.6 

+  133.0 

1000 

289  .5 

90313 

8.3 

+  23.0 

292  .6 

90228 

9.2 

+  49.2 

800 

291  .0 

92460 

8.1 

+  19.7 

294  .8 

92355 

8.7 

+  49.2 

600 

292  .5 

94656 

7.9 

+  18.9 

296  .9 

94510 

8.3 

+  49.9 

400 

294  .5 

96886 

7.6 

+  16.2 

299  .0 

96703 

7.9 

+  52.2 

200 

296  .2 

99154 

6.0 

+  23.4 

300  .5 

98930 

7.0 

+  52.2 

000 

297  .7 

101459 

4.0 

301  .3 

101214 

5.0 

6  P.I 

I. 

10  F 

.M. 

2500 

281°.  5 

75458 

8.0 

+159.3 

280°.  8 

75466 

5.0 

+  142.0 

2000 

286  .0 

80142 

9.0 

+  156.5 

284  .3 

80170 

6.0 

+117.6 

1500 

290  .0 

85038 

9.0 

+  145.6 

287  .1 

85102 

6.0 

+  88.5 

1000 

293  .5 

90166 

8.0 

+  51.9 

290  .7 

90294 

6.0 

+  34.9 

800 

296  .0 

92278 

8.0 

+  58.9 

292  .2 

92432 

5.0 

+  28.0 

600 

297  .3 

94426 

7.0 

+  51.6 

294  .2 

94606 

5.0 

+  29.0 

400 

298  .0 

96636 

6.0 

+  42.1 

295  .1 

96835 

4.0 

+  19.9 

200 

298  .6 

98867 

5.0 

+  31.6 

295  .7 

99095 

3.0 

+  3.2 

000 

299  .0 

101149 

4.0 

294  .3 

101414 

2.0 

from  the  descending  air  to  the  ground,  and  an  inversion  of  tem- 
perature under  ordinary  circumstances.  The  friction  and  the 
earth's  deflection  have  very  little  influence  on  the  temperature 
and  pressure  conditions,  and  the  circulation  cannot  be  studied 
by  itself  until  the  radiation  or  heating  terms  have  been  eliminated. 


104 


THERMODYNAMIC  METEOROLOGY 


TABLE  25 

THE  CORRESPONDING  VALUES  OF  Bc=P/g0  Pm  ON  THE  LEVEL  2  =  000 
METERS  ARE  Now  GIVEN,  AND  COMPARED  WITH  THE  OBSERVED  VALUES 
OF  #  BY  MEANS  OF  THE  DIFFERENCES 


Surface 

2A.M. 

6A.M. 

10  A.M. 

2  P.M. 

6P.M. 

10  P.M. 

Mean 

Bc 
Bo 
A  BC 

760.20 
760.16 
+0  08 

760.88 
760.75 
+0  76 

761.03 
760.98 
+0  91 

759.18 
759.52 
-0  94 

758.70 
758.71 
—  1  42 

760.70 
760.59 

+0  58 

760.12 
760.12 

A  BO 

+0  04 

+0  63 

+0  86 

-0  60 

—  1  41 

+0  47 

These  curves  can  be  reduced  to  the  harmonics  if  desired.  The 
value  of  the  radiation  exponent  is  a  =  3.82  throughout  the 
twenty-four  hours. 

The  Thermodynamic  Structure  of  Cyclones  and  Anticyclones 

There  has  been  much  speculation  regarding  the  forces  that 
generate  the  powerful  circulations  in  storms  known  by  the  name 
of  cyclones  and  anticyclones,  or  low-pressure  and  high-pressure 
areas  respectively.  These  will  be  more  fully  mentioned  in  the 
chapter  on  Dynamic  Meteorology,  but  here  we  proceed  to  apply 
the  principles  just  illustrated  in  the  diurnal  convection.  From 
numerous  kite  and  balloon  ascensions  in  all  parts  of  these  local 
circulations,  it  has  been  learned  what  is  the  usual  distribution 
of  the  temperature,  and  from  the  cloud  observations  what  is  the 
direction  and  velocity  of  the  wind  motion  or  " vector"  in  all 
areas,  and  all  altitudes  up  to  at  least  10,000  meters.  Compare 
the  International  Cloud  Report,  1898,  the  Monthly  Weather 
Review,  January  to  July,  1902,  April  to  June,  1904,  January  to 
August,  1906,  October,  1907,  to  February,  1909,  also  the  daily 
Weather  Synoptic  Charts,  for  numerous  studies  and  details. 
From  these  data  we  have  selected  the  temperatures  T,  and  veloc- 
ities q,  given  in  Tables  26,  25,  and  Figs.  11,  12.  The  computed 
values  of  the  pressure  P,  and  the  free  heat  (Qi  -  Qo)  are  given 
in  the  same  tables  and  figures,  while  the  other  thermodynamic 
data  are  summarized  in  Bulletin  No.  3,  0.  M.  A.  Only  three 
diagrams  are  extracted  from  the  tables,  but  these  are  enough  to 


6000  meterp         High  Area 


Low  Area 


266 


2000  meters 


260 


284  288         292 

Full  lines  =  Temperatures   T 
Dotted  lines = Variations  of  Heat      (Q,—  Q0) 
FIG.    ii.     The  temperatures  and  heat  variations  in  high  and  low  areas 


High  Area 


5000  meter? 


FIG.    12.     Pressures  and  wind  velocities  in  high  and  low  areas 


THERMODYNAMIC    STRUCTURE    OF    CYCLONES 

TABLE  26 
SOME  VALUES  OF  T.  P.  q(Ql-  Q0)  IN  CYCLONES 

TEMPERATURE  T 


107 


000 

500 

1000 

1500 

2000 

2500 

3000 

4000 

5000 

5.      760 
S.     750 
5.      740 
C.     735 

290.0 

287.5 
284.0 
280.0 

289.0 
286.4 
282.6 
279.0 

288.0 
284.5 
280.8 
277.5 

286.3 
282.7 
279.0 
275.5 

284.3 
280.4 
276.8 
273.2 

282.1 
278.0 
274.4 
271.0 

279.8 
276.0 
271.6 
269.0 

274.5 
270.5 
266.0 
263.0 

268.0 
264.0 
260.0 
257.0 

E.     760 
E.     750 
E.     740 

278.0 
281.0 
283.5 

276.6 
279.3 
281.4 

274.8 
277.0 
279.1 

272.4 
274.6 
276.6 

269.7 
272.0 
274.0 

267.0 
269.5 
271.0 

264.5 
267.0 
268.0 

259.5 
261.7 
263.0 

255.0 
256.0 
257.0 

N.     750  I 
N.     750  II 

N.     740 

266.0 
272.5 
277.0 

265.8 
271.2 
276.0 

264.5 
269.7 
274.8 

263.0 
268.0 
273.0 

261.5 
266.0 
270.8 

259.5 
263.9 
268.5 

257.0 
261.7 
266.0 

250.0 
256.0 
260.5 

243.0 
249.0 
254.0 

W.    760 
W.    750 
W.    740 

268.0 
273.0 
277.0 

267.5 
272.6 
276.0 

266.8 
272.0 
275.0 

266.0 
271.0 
273.7 

265.0 
269.8 
272.0 

264.0 
268.0 
270.0 

262.0 
266.0 
268.0 

258.0 
261.0 
263.0 

253.0 
255.0 
256.5 

PRESSURE  P 


S.   760 

101322 

95518 

90010 

84814 

79893 

75208 

70778 

62566 

55161 

5.   750 

99988 

94194 

88724 

83548 

78628 

73953 

69540 

61367 

54000 

S.   740 

98655 

92873 

87418 

82230 

77340 

72683 

68278 

60127 

52801 

C.  735 

97990 

92158 

86672 

81482 

76565 

71908 

67498 

59366 

52051 

E.  760 

101322 

95275 

89562 

84140 

78997 

74124 

69515 

61016 

53424 

E.  750 

99988 

94064 

88462 

83150 

78114 

73338 

68812 

60467 

52993 

E.  740 

98655 

92864 

87380 

82166 

77223 

72532 

68072 

59861 

52486 

N.  750  I 

99988 

93706 

87884 

82372 

77165 

72273 

67641 

59114 

51465 

2V.  750  II 

99988 

93890 

88142 

82708 

77587 

72747 

68163 

59737 

52179 

N.  740 

98655 

92747 

87176 

81896 

76915 

72187 

67720 

59481 

52080 

W.  760 

101322 

95067 

89220 

83654 

78460 

73532 

68911 

60404 

52860 

W.  750 

99988  93983 

88252 

82868 

77790 

72997 

68477 

60150 

52690 

W.  740 

98655 

92745 

87156 

81890 

76930 

72223 

67779 

59603 

52256 

VELOCITY  q 


S.  760 

4.1 

6.5 

9.0 

13.9 

21.0 

26.5 

27.0 

24.0 

23.0 

5.   750 

4.9 

9.0 

12.0 

15.4 

21.2 

27.0 

26.0 

24.0 

26.0 

S.   740 

6.5 

11.0 

14.0 

16.9 

22.8 

30.0 

34.0 

34.0 

33.0 

C.  735 

7.0 

11.0 

15.0 

19.0 

30.0 

40.0 

42.0 

40.0 

39.0 

E.  760 

5.2 

8.6 

12.4 

16.0 

19.8 

23.0 

24.0 

25.0 

26.0 

E.  750 

5.0 

9.0 

13.0 

17.0 

22.0 

26.0 

27.0 

27.0 

27.0 

E.  740 

5.0 

7.5 

11.0 

15.0 

22.0 

29.0 

29.0 

29.0 

28.0 

TV.  750  I 

3.5 

5.0 

5.8 

6.0 

7.0 

7.5 

8.0 

8.0 

7.5 

N.  750  II 

4.5 

6.0 

8.0 

11.0 

14.5 

18.0 

20.5 

24.5 

25.0 

TV.  740 

6.0 

8.0 

11.0 

15.0 

18.0 

22.0 

26.0 

31.0 

33.0 

W.  760 

5.0 

8.0 

12.0 

17.0 

22.0 

26.0 

27.0 

28.0 

31.0 

W.  750 

6.0 

9.0 

13.0    19.0 

24.0 

27.0 

28.0 

30.0 

33.0 

W.  740 

6.0 

10.0 

14.0    20.0 

25.0 

29.0 

32.0 

34.0 

37.0 

1 

108 


THERMODYNAMIC   METEOROLOGY 


FREE  HEAT  (Qi  -  Qo) 


5.  760 

-45.9 

-118.4 

-209.7 

-333.0 

-386.3 

-313.4 

-789.6 

-872.3 

5.  750 

-61.2 

-124.5 

-189.3 

-297.2 

-374.2 

-253.8 

-631.9 

-861.6 

5.  740 

-69.2 

-125.2 

-186.0 

-316.2 

-425.8 

-403.3 

-659.0 

-764.7 

C.  735 

... 

-70.5 

-150.6 

-221.4 

-465.6 

-597.1 

-377.6 

-625.6 

-810.9 

E.  760 

-54.5 

-129.3 

-189.8 

-248.3 

-282.7 

-283.5 

-671.9 

-851.9 

E.  750 

-55.7 

-122.8 

-184.1 

-263.8 

-303.7 

-276.4 

-621.4 

-773.4 

E.  740 

-55.0 

-104.1 

-167.4 

-286.0 

-377.2 

-232.2 

-571.8 

-695.6 

N.  750  I 

-50.1 

-124.3 

-184.7 

-251.3 

-304.7 

-352.3 

-791.3 

-890.2 

N.  750  II 

-40.6 

-109.8 

-183.5 

-254.0 

-316.8 

-356.8 

-828.4 

-874.3 

2V.  740 

-48.6 

-130.3 

-213.5 

-261.7 

-338.5 

-398.1 

-864.0 

-919  6 

W.  760 

-59.7 

-158.1 

-260.1 

-360.9 

-427.8 

-417.6 

-960.4 

-1-204  .  0 

W.  750 

-62.5 

-167.1 

-286.6 

-364.4 

-392.4 

-394.8 

-925.4 

-1110.3 

JF.  740 

-66.7 

-151.7 

-271.5 

-341.0 

-389.2 

-417.7 

-860.9 

-1041.4 

TABLE  27 
SOME  VALUES  OF  T.  P.  q  (Ql  —  <20)  IN  ANTICYCLONES 

TEMPERATURE  T 


A 

C       B 

000 

500 

1000 

1500 

2000 

2500 

3000 

4000 

5000 

14 
15 
16 
17 

S.     770  I 
S.     770  II 
S.     775 
C.     777 

278.0 
273.5 
272.0 
272.0 

277.4 
273.0 
271.0 
271.0 

276.0 
272.0 
269.5 
269.2 

274.6 
270.4 
267.8 
267.0 

273.0 
268.7 
266.0 
264.5 

271.0 
267.0 
264.0 
262.0 

268.6 
265.0 
262.0 
259.5 

264.0 
260.0 
256.5 
254  .  5 

259.0 
254.0 
250.0 
248.0 

11  =  18 
19 
20 

E.     760 
E.     770 

E.     775 

268.0 
264.0 
266.0 

267.5 
263.3 
265.0 

266.8 
262.7 
264.0 

266.0 
262.0 
262.9 

265.0 
261.2 
261.5 

264.0 
259.7 
259.8 

262.0 
258.0 
258.0 

258.0 
254.0 
254.0 

253.0 
249.5 
249.0 

21 
22 
23 

N.    760 
N.     770 

N.     775 

271.0 
272.0 
272.0 

269.8 
271.4 
271.0 

268.2 
270.3 
269.5 

266.3 
269.1 
267.6 

264.2 
267.6 
265.5 

262.0 
265.4 
263.1 

259.4 
262.6 
260.4 

254.0 
257.0 
255.0 

248.0 
251.0 
249.5 

24 
25 
26 

W.    760 
W.    770 

W.    775 

282.0 
278.0 
274.0 

281.2 
277.1 
273.3 

280.3 
276.0 
271.8 

279.1 

274.7 
270.0 

277.7 
273.0 
268.0 

276.0 
271.0 
265.9 

273.8 
268.5 
263.7 

268.0 
263.0 
258.0 

262.0 
257.0 
252.0 

PRESSURE  P 


14 

5.  770  I 

102655 

96583 

90794 

85330 

80146 

75292 

70677 

62160 

54551 

15 

5.  770  II 

102655 

96594 

90550 

85028 

79805   74865 

70220 

61644 

53970 

16 

S.  775 

103322 

97020 

91080 

85466 

80164  i  75157 

70435 

61741 

53949 

17 

C.  777 

103588 

97270 

91316 

85670 

80343 

75289 

70520 

61740 

53888 

18 

E.  760 

101322 

95067 

89220 

83654 

78460 

73532 

68911 

60404 

52860 

19 

E.  770 

102655 

96232 

90184 

84492 

79144 

74122 

69392 

60711 

53016 

20 

E.  775 

103322 

96906 

90818 

85126 

79753 

74698 

69928 

61187 

53419 

21 

N.  760 

101322 

95110 

89266 

83736 

78512 

73577 

68913 

60321 

52646 

22 

N.  770 

102655 

96394 

90676 

85088 

79857 

74893 

70197 

61549 

53803 

23 

N.  775 

103322 

97020 

91080 

85472 

80153   75142 

70395 

61656 

53845 

24 

W.  760 

101322 

95376 

89716 

84410 

79394 

74638 

70136 

61824 

54349 

25 

W.  770 

102655 

96563 

90772 

85312 

80146  !  75273 

70648 

62111 

54476 

26 

W.  775 

103322 

97073 

91176 

85596 

80327 

75357 

70647 

61974 

54203 

THERMODYNAMIC    STRUCTURE    OF    CYCLONES 
VELOCITY  q 


109 


14 

5.  770  I 

5.0 

7.0 

10.0 

12.0 

16.0 

19.0 

20.0 

21.0 

22.0 

15 

5.  770  II 

5.0 

7.0 

8.0 

9.0 

12.0 

15.0 

16.0 

17.0 

18.0 

16 

5.  775 

3.0 

5.0 

7.0 

8.0 

9.0 

11.0 

12.0 

14.0 

16.0 

17 

C.  777 

4.0 

5.0 

6.0 

8.0 

9.0 

10.0 

11.0 

12.0 

14.0 

18 

E.  760 

5.0 

8.0 

12.0 

17.0 

22.0 

26.0 

27.0 

28.0 

31.0 

19 

E.  770 

5.0 

7.0 

10.0 

14.0 

19.0 

23.0 

25.0 

26.0 

28.0 

20 

E.  775 

4.0 

6.0 

9.0 

12.0 

16.0 

19.0 

20.0 

22.0 

24.0 

21 

N.  760 

5.0 

8.0 

11.0 

14.0 

18.0 

22.0 

24.0 

26.0 

27.0 

22 

2V.  770 

6.0 

9.0 

13.0 

16.0 

20  0 

23.0 

26.0 

28.0 

30.0 

23 

N.  775 

4.0 

7.0 

10.0 

13.0 

17.0 

21.0 

23.0 

24.0 

26.0 

24 

Wr.  760 

5.0 

8.0 

10.0 

13.0 

14.0 

15.0 

16.0 

17.0 

18.0 

25 

PF.  770 

5.0 

7.0 

9.0 

11.0 

12.0 

13.0 

14.0 

15.0 

16.0 

26 

W.  775 

4.0 

6.0 

8.0 

10.0 

12.0 

12.5 

13.0 

14.0 

15.0 

FREE  HEAT  (Qi  -  Qo) 


14 

5.  770  I 

—49  7 

—131  4 

-189  0 

—282  0 

-337  1 

—350  6 

—804  4 

—977  1 

15 

5.  770  II 

—51  8 

—121  5 

-186  1 

—266  1 

—331  1 

—359  0 

—837  1 

—988  0 

16 

5.  775 

-43.3 

-113.2 

-167.7 

-224.8 

-289.5 

-332.2 

-794.6 

-934.0 

17 

C.  777 

-39.8 

-104.0 

-164.6 

-204.8 

-248.9 

-293.1 

-706.1 

-868.1 

18 
19 

E.  760 
E.  770 

.**.*• 

-59.7 
—51  0 

-158.1 
—142  8 

-260.1 

-242  8 

-360.9 
—352  0 

-427.8 
—420  2 

-417.6 
—442  4 

-960.4 
—972  2 

-1204.5 
-1194  7 

20 

E.  775 

-45.9 

-130.2 

-209.3 

-299.6 

-356.1 

-379.1 

-918.7 

-1109.7 

21 
22 
23 

TV.  760 
TV.  770 
N.  775 

-53.2 
-61.3 
—51  8 

-125.7 
-154.6 
—  126.7 

-191.5 
-219.9 
—192  9 

-270.1 
-311.0 
—270  2 

-335.3 
-356.9 
—333  5 

-346.0 
-408.2 
—343  4 

-770.4 
-834.3 
—740  4 

-895.7 
-980.2 
—923  1 

24 

W.  760 

—55  2 

—  123  9 

—206  7 

—246  8 

—304  4 

—355  1 

-808  8 

-942  4 

25 

W.  770 

-47.2 

-119.8 

-188.3 

-238.6 

-292.2 

-338.8 

-780.6 

-921.9 

26 

W.  775 

-47.7 

-119.5 

—  181.1 

-238.1 

-272.5 

-321.0 

-762.3 

-903.0 

indicate  clearly  the  principles  that  are  involved  in  their  structure. 
Continuing  the  computations  to  higher  levels,  it  is  found  that  the 
temperature  lines  or  isotherms,  the  pressure  lines  or  isobars,  and 
the  velocity  lines  or  vectors,  coincide  at  every  point  in  direction. 
In  the  lower  levels  these  lines  cross  each  other  at  various  angles 
in  the  areas  marked  1  to  26  on  the  sea  level  of  Fig.  12,  which 
shows  the  order  of  the  computations:  (1)  The  upper  undisturbed 
circulation,  where  T.  P.  q  coincide,  belongs  to  the  general 
circulation,  and  (2)  the  lower  disturbed  circulation,  where 
T.  P.  q  do  not  coincide,  to  the  combined  general  and  local  circula- 
tion; (3)  the  purely  local  circulation,  the  cyclone  and  anti- 
cyclone proper,  can  be  separated  from  the  second  by  vector 
composition,  since  (2)  is  the  resultant  of  (l)  and  (3).  It  is 
found  that  the  disturbing  circulation  (3)  is  similar  in  configuration 


110  THERMODYNAMIC  METEOROLOGY 

to  that  at  sea  level  up  to  10,000  meters,  if  it  penetrates  the  general 
eastward  drift  so  high,  that  it  usually  increases  in  intensity  to 
the  3,000-meter  level,  and  then  gradually  dies  out  as  the  head  is 
stripped  away  in  the  rapidly  flowing  upper  currents.  There 
is  no  evidence  of  a  change  in  the  type  of  the  circulation,  and 
therefore  the  physical  origin  of  the  structure  of  a  cyclone  is  the 
same  throughout.  In  the  International  Cloud  Report  it  was 
shown  that  a  series  of  warm  currents  from  the  south  interlock 
alternately  with  another  series  of  cold  currents  from  the  north, 
in  the  United  States  and  adjacent  regions,  and  that  these  local 
circulations  are  the  mixing  regions  where  the  interchange  of 
the  temperature  goes  on  toward  a  thermodynamic  equilibrium 
under  the  force  of  gravitation.  Fig.  11  shows  the  temperature 
distribution  T  and  the  free-heat  distribution  (Qi  —  Qo).  The 
temperature  is  deflected  to  the  south  on  the  east  side  of  the 
high  area,  and  to  the  north  on  the  east  side  of  the  low  area.  The 
deflections  of  the  isotherms  diminish  with  the  altitude  and 
finally  disappear  as  these  disturbing  currents  diminish  in  strength. 
There  are  no  cold-center  anticyclones,  and  no  warm-center 
cyclones,  as  has  been  assumed  in  many  theoretical  discussions. 
The  distribution  of  (Qi  —  Q0)  is  in  elliptical  figures  whose  centers 
are  on  the  border  of  the  high  and  low  areas,  and  they  show  where 
the  exchange  of  heat  is  going  on  most  vigorously.  The  radiation 
heat  increases  with  the  height  in  consequence  of  the  general 
radiation  of  the  atmosphere  increasing  upward.  It  would  be 
well  to  separate  the  purely  local  (Qi  —  Q0)  from  the  general  as 
can  be  done  by  computation.  The  pressures  of  Fig.  12  depend 
entirely  upon  the  temperature  assigned  to  the  several  areas,  and 
not  upon  the  circulation,  the  deflecting  force,  the  centrifugal 
force,  or  the  friction,  or  any  other  minor  condition.  The  air 
column,  though  temporary  in  position  at  a  given  instant,  presses 
upon  the  level  of  computation,  in  consequence  of  the  air  masses 
which  are  determined  only  by  the  density,  since  this  in  turn  is 
a  function  of  the  temperature.  The  given  temperature  structure 
must  be  continuously  renewed  by  circulation  of  air  from  the  warm 
and  cold  regions,  or  else  the  gravitation  would  soon  flatten 
down  all  the  disturbed  temperature  and  pressure  levels.  The 


WARM,    COLD,    AND    LEAKAGE    CURRENTS  111 

ultimate  source  of  heat  is  the  sun's  insolation,  chiefly  on  the 
tropics,  and  radiation  in  general  from  the  atmosphere.  The 
tropics  are  the  boiler  and  the  polar  regions  the  condenser  of 
the  thermal  engine,  and  the  cyclones  and  anticyclones  are  the 
working  machinery  of  motion.  The  general  circulation  depends 
upon  the  heat  of  the  tropics,  with  westward  drift  to  the  south 
of  33°  latitude,  and  eastward  drift  in  middle  latitudes  from  33° 
to  66°  latitude,  that  in  the  polar  zone  being  irregular.  There 
are,  however,  centers  of  general  action  along  the  high-pressure 
belt  separating  the  westward  from  the  eastward  drifts,  such 
that  there  are  leakage  currents  from  one  of  these  zones  to  the 
other.  There  are  such  centers  of  action  over  the  tropic  north 
and  south  Atlantic  and  Pacific  Oceans,  those  in  the  same  hemi- 
sphere being  broken  through  by  the  western  and  eastern  conti- 
nents. Such  leakage  currents  flow  northward  from  the  Gulf  of 
Mexico  over  the  United  States,  and  from  the  north  Pacific  upon 
the  northwestern  States  in  a  southward  direction.  The  inter- 
flowing of  these  two  series  of  warm  and  cold  currents  upon  the 
United  States  and  Canada  is  the  immediate  cause  of  the  num- 
erous cyclones  and  anticyclones  that  wander  eastward  over 
this  region.  Forecasts  are  made  of  the  probable  detailed  action 
of  the  weather  conditions  in  all  areas,  as  learned  by  experience 
with  the  types  that  these  local  circulations  assume.  The 
operation  of  the  several  elements,  temperature,  pressure,  wind, 
and  precipitation,  is  very  complex  and  irregular,  so  that  practical 
forecasts  are  difficult  and  uncertain,  except  in  the  cases  of 
vigorously  developed  storms,  which  move  along  paths  quite 
well  determined  by  the  pressure  and  temperature  distributions. 
It  should  be  noted  in  Fig.  12  that  there  is  a  saddle  of  higher 
pressure  to  the  north  of  the  center  in  low  areas,  and  to  the  south 
of  high  areas  there  is  a  saddle  of  low  pressure.  These  gradually 
diminish  with  the  height  and  usually  disappear  above  the  3,000- 
meter  level.  It  can  be  seen  that  the  winds  on  the  sea  level 
generally  blow  out  of  a  high  area  into  a  low  area,  by  curves 
having  reversed  spiral  forms  crossing  the  isobars  at  angles 
varying  with  the  place.  The  winds  flow  more  closely  along 
the  isobars  at  higher  levels,  as  stated,  and  from  5,000  meters  to 


112  THERMODYNAMIC   METEOROLOGY 

10,000  meters  it  would  be  safe  to  draw  the  isobars  and  isotherms 
from  the  wind  directions  as  observed  in  the  high  clouds.  It 
follows  that  high-level  pressure  and  temperature  charts  are 
the  true  indicators  of  the  general  movement  of  storms  across 
the  continent,  because  they  show  the  direction  of  the  eastward 
drift  when  the  isobars  on  the  sea  level  do  not  clearly  indicate  it. 
Such  charts  were  prepared  by  the  author  for  the  sea  level,  the 
3,500-foot  plane,  and  the  10,000-foot  plane  for  the  United  States, 
and  they  proved  to  be  most  instructive  for  the  public  forecast 
service. 

Attention  is  called  to  the  fact  that  the  isobars  are  all  marked 
in  the  notation  of  Table  1,  units  of  force  B  =  P/100,  and  the 
isotherms  in  absolute  temperatures  T.  Thus,  we  have  for  the 
pressure : 

D  Tt  Mercury 

f  D  mm. 

103322  1033.2  775.0 

102655  1026.5  770.0 

101322  1013.2  760.0 

99988  999.9  750.0 

98655  986.6  740.0 

97990  979.9  735.0 

P  is  the  dynamic  pressure  in  the  M.  K.  S.  System;  B  is 
this  pressure  divided  by  100  for  practical  use;  mm.  is  the  milli- 
meters of  mercury  of  a  barometer.  B  and  mm.  are  related 
very  closely  in  the  ratio  4  to  3.  Hence,  by  making  the  scale  of 
a  barometer  in  divisions  each  three-fourths  of  a  millimeter,  it 
would  be  only  necessary  to  multiply  the  reading  by  100  to  obtain 
the  dynamic  pressures  useful  in  all  computations  on  the  dynamics 
and  thermodynamics  of  the  atmosphere.  This  change  in  units 
is  so  simple,  and  so  far-reaching  in  its  beneficial  results,  that  it 
is  strongly  recommended  to  meteorologists.  Generally,  a  full 
set  of  Tables  should  be  constructed  to  supersede  the  mercurial 
British  and  the  Metric  systems  now  in  use. 

Further  attention  is  called  to  the  fact  that  these  computa- 
tions fully  satisfy  equation  (196)  in  a  non-adiabatic  atmosphere, 
and  that,  therefore,  the  author's  theory  of  the  non-asymmetric 
cyclone  and  anticyclone,  due  to  interflowing  currents  at  different 
temperatures  subject  to  the  attraction  of  gravitation  at  every 


PLANETARY   CIRCULATION  AND   RADIATION  113 

point,  is  fully  verified,  since   the  computed  and  the    observed 
values  are  in  agreement. 

The  Planetary  Circulation  and  Radiation.     The  Observations  of 
Temperature  and  Velocity 

The  greatest  difficulty  in  discussing  the  problems  of  the 
planetary  circulation  and  radiation  consists  in  determining  the 
proper  temperatures  and  velocities  of  the  circulation  in  all 
latitudes  from  the  equator  to  the  pole,  and  at  all  altitudes  from 
the  surface  up  to  the  practical  limit  of  the  balloon  ascensions, 
as  30,000  meters.  The  number  of  available  observations  is  very 
limited  throughout  the  tropics,  they  are  lacking  entirely  in 
the  arctic  zone,  and  above  14,000  meters  in  the  isothermal 
region  they  are  insufficient  for  our  purposes.  In  spite  of  these 
difficulties  it  has  been  thought  proper  to  execute  the  extensive 
computations,  for  the  sake  of  the  general  instruction  regarding 
various  unsolved  problems  of  meteorology,  which  depend  upon 
such  data.  There  are  several  accessible  reports  and  compilations 
on  the  results  of  balloon  ascensions,  and  we  utilize  them  without 
further  references:  Rykachef  for  Russia,  Dines  for  England, 
Teisserenc  de  Bort  for  France,  Wegener  for  Germany,  Rotch  for 
St-  Louis,  Teisserenc  de  Bort  and  Rotch  for  the  Atlantic  Ocean, 
Berson  for  Victoria  Nyanza  and  East  Africa.  Table  27 
contains  a  summary  of  the  original  mean  observations  arranged 
according  to  the  latitude,  and  Table  30  contains  the  adopted 
temperature  system,  which  fairly  represents  this  type  of  distribu- 
tion. An  inspection  of  these  original  temperatures  presents  a 
great  difficulty  when  they  are  compared  with  the  wind  velocities 
and  directions  in  the  tropics.  It  is  seen  that  there  is  a  decrease 
of  temperature  in  the  convectional  region  from  the  equator  to 
the  pole,  except  in  the  low  levels  of  the  tropics,  as  indicated 
in  Fig.  13,  Case  II.  When  the  temperature  rises  towards  the 
pole  there  is  westward  wind,  as  in  the  trades  of  the  tropics; 
when  the  temperature  falls  toward  the  pole  there  is  eastward 
drift,  as  in  the  temperate  zones.  This  was  first  developed  by 
Bigelow,  1904,  and  confirmed  by  De  Bort  and  Rotch  in  their 
report,  1909,  thus  establishing  a  fundamental  property  of  all 


114 


THERMODYNAMIC   METEOROLOGY 


atmospheric  motions  as  indicated  by  Helmholtz.  The  trades 
blow  steadily  westward  at  a  moderate  velocity,  while  the  east- 
ward drift  reaches  a  mean  velocity  of  about  35  meters  per  second 


z 

Height 

in 
Meters 

18000 

17000 

16000 

15000 

14000 

13000 

12000 

11000 

10000 

9000 

8000 

7000 

6000 

5000 

4000 

3000 

2000 

1000 

000 


Case  II  High  Temperatures 
in  the  Isothermal  Region 


90°  80°  70°  60°  50°  40°  30°  20°  10°  0° 


Case  I  Low  Temperatures 
in  the  Isothermal  Region 


70°  60°  50°  40°  30°  20°  10°  Oc 


FIG.   13.     Two  typical  cases  of  the  observed  temperatures  in  the  earth's 
atmosphere  up  to   19,000  meters 

in  middle  latitudes,  where  the  temperature  falls  rapidly  toward 
the  pole  at  high  elevations. 

These  cases  illustrate  the  movement  of  a  temperature  maxi- 
mum from  the  tropics  into  the  temperate  zones  of  the  isothermal 
region. 

The  general  questions  of  temperature  are  greatly  complicated 
by  the  necessity  of  adapting  them  to  the  observed  velocities, 
and  for  those  the  observations  are  too  limited  in  number  in  the 
isothermal  region  and  in  the  arctic  zone  to  be  decisive.  Table 
33  contains  one  system  of  velocities,  which  conform  to  the 
adopted  temperatures  of  Table  30.  The  velocities  are 
directed  westward  in  the  convection  region  of  the  tropics,  with 
maximum  on  the  4,000-meter  level,  minimum  at  the  13,000- 
meter  level,  and  a  region  of  alternately  westward  and  eastward 
velocities  above  that  level,  except  immediately  over  the  equator, 
where  the  wind  is  steadily  westward.  In  the  latitudes  on  the 
poleward  side  of  the  high-pressure  belt,  which  is  in  latitude  30°, 


PLANETARY   CIRCULATION   AND   RADIATION 


115 


TABLE  27 

THE  MEAN  OBSERVED  TEMPERATURES  IN  THE  ATMOSPHERE  FROM  BALLOON 
ASCENSIONS  ARRANGED  IN  THE  ORDER  OF  LATITUDE 


Russia 

Eng- 
land 

Ger- 

many 

France 

St. 
Louis 

Atlantic  Ocean  Tropics 

Viet. 
Nyanza 

Lati- 

tude 

56° 

53° 

51° 

49° 

39° 

35° 

25°           15° 

5° 

0° 

Num- 

ber 

143 

200 

380 

581 

23 

12 

6 

8 

6 

12 

19000 

18000 

190  5 

17000 

197  1 

16000 

202  6 

15000 

213  3 

206  8 

14000 

219.1 

218.6 

218.9 

212.2 

211.2 

210.7 

210.0 

214.1 

210.8 

13000 

.... 

219.3 

218.5 

218.6 

214.2 

214.6 

212.6 

216.2 

218.1 

216.0 

12000 

218.3 

219.6 

218.8 

217.8 

216.7 

219.4 

217.4 

223.6 

224.7 

222.6 

11000 

217.7 

219.4 

220.2 

219.0 

221.0 

225.8 

224.7 

231.8 

231.8 

231.4 

10000 

218.9 

223.1 

223.4 

223.7 

226.2 

233.7 

233.2 

240.2 

239.3 

238.9 

9000 

224.3 

228.4 

228.6 

229  .5      232  .  9 

242.0 

241.8 

248.2 

247.2 

246.1 

8000 

231.4 

235.2 

235.0 

236.8      239.8 

250.0 

249.6 

255.5 

254.6 

250.7 

7000 

239.1 

241.8 

242.2 

244.0      248.5 

257.6 

256.6 

261.8 

261.4 

258.0 

6000 

246.4 

248.8 

249.3 

251.1 

256.1 

264.4 

263.0 

267.6 

267.3 

263.4 

5000 

253.1 

255.4 

256.1 

257.6 

262.7 

270.5 

268.8 

272.8 

272.1 

269.2 

4000 

259.3 

261.7 

262.3 

263.6 

268.6 

275.8 

274.6 

277.9 

277.2 

274.7 

3000 

265.3 

268.8 

268.0 

269.0 

273.5 

280.9 

279.3 

282.8 

282.0 

280.8 

2000 

269.9 

272.5 

273.1 

273.7 

277.8 

285.4 

283.9 

287.5 

286.7 

288.4 

1000 

274.3 

2,77.1 

277.6 

278.3 

281.1 

290.9 

288.7 

292.2 

292.3 

296.2 

000 

277.1 

281.3 

282.9 

282.5 

285.9 

298.9 

296.2 

298.4 

300.8 



as  given  in  Table  29  for  all  elevations,  the  wind  is  eastward, 
increasing  from  the  surface  to  a  maximum  on  the  9,000-10,000- 
meter  level,  where  it  suddenly  falls  in  velocity,  and  prevails 
eastward  or  variable  throughout  the  isothermal  region.  These 
velocity  conditions  conform  to  Bigelow's  observations  at  Wash- 
ington, D.  C.,  1896-97,  International  Cloud  Report,  Charts 
11,  14,  and  Monthly  Weather  Review,  April,  May,  June,  1904. 
The  reports  of  the  International  Committee  show  that  for  the 
hemisphere  at  large,  these  westward  and  eastward  circulations 
tend  to  concentrate  about  "centers  of  action,"  wherein  the 
continuity  of  the  high-pressure  belt  around  the  globe  in  longitude 
is  broken  up  into  sections,  one  over  the  Atlantic  Ocean  in  each 
hemisphere,  and  another  over  the  Pacific  Ocean  in  each  hemi- 
sphere. This  subdivision  is  due  to  the  mutual  influence  of 


116 


THERMODYNAMIC  METEOROLOGY 


oceans  and  continents  through  the  induced  temperatures,  and 
the  configurations  of  the  great  currents  of  the  general  circulation 
depending  upon  them.  The  high-pressure  belt  is  itself  produced, 
in  that  latitude,  by  the  downflow  of  air  which  has  originally 
risen  in  the  tropics;  the  segregation  is  accompanied  by  low-level 
"  leakage  currents  "  from  the  tropics  to  the  temperate  zones,  which 
form  the  warm  parts  of  cyclones  and  anticyclones.  The  cold 
streams  from  the  polar  zones  meet  these  warm  currents  in  mid- 
latitudes,  and  their  interaction  produces  the  local  circulations  of 
storms,  under  the  force  of  gravitation  acting  on  warm  and  cold 
masses  in  contact  with  each  other. 

For  our  special  purpose  in  this  connection,  there  has  been 
much  discussion  regarding  the  existence  of  the  " antitrade" 
winds  blowing  eastward  in  the  upper  levels,  as  above  12,000 
meters.  The  observational  data  are  themselves  conflicting,  but 
this  points  to  a  very  important  feature  in  the  theory  of  the 
planetary  circulation.  Collecting  some  of  the  data  for  very 
high  altitudes,  9,000  to  17,000  meters,  we  have  the  following 
typical  exhibit: 

TABLE  28 

THE  NUMBER  OF  WINDS  FROM  EIGHT  COMPASS  POINTS,  IN  DIFFERENT 
LATITUDES,  AT  9,000  TO  17,000  METERS  ALTITUDE 


Station 

N. 

N.E. 

E. 

S.E. 

[s. 

S.W. 

w. 

N.W. 

To'l 

Lat. 

Lindenburg  

18 

17 

14 

4 

1 

16 

2 

q 

81 

52° 

Ponta  Delgada  . 

6 

16 

8 

4 

1 

10 

1? 

q 

66 

37°.7 

and  Madeira. 

32°.6 

Teneriffe  

3 

0 

?, 

?, 

3 

?,?, 

1 

g 

38 

28°.5 

St.  Vincent  

0 

?, 

13 

4 

5 

2 

0 

0 

26 

17° 

Victoria  Nyanza  
Ascension  

5 
1 

10 

o 

36 
5 

10 
0 

2 
0 

4 
0 

4 
0 

10 

8 

81 
14 

0° 

-8° 

It  is  evident  that  there  is  an  alternation  of  wind  direction 
between  N.E.  and  S.W.  in  all  latitudes,  but  that  the  westward 
wind  prevails  over  the  equator,  and  to  some  extent  predominates 
in  all  latitudes  to  50°  or  60°.  This  can  only  mean  that  in  the 
isothermal  region  the  temperatures  increase  from  the  equator 


PLANETARY    CIRCULATION   AND    RADIATION  117 

to  that  latitude,  as  on  Fig.  13,  Case  II,  "High  Temperatures 
in  the  Isothermal  Region";  for  all  the  wind  directions  blowing 
eastward,  the  opposite  temperature  gradient  must  occur,  of 
fall  from  the  equator  toward  the  pole,  as  in  Case  I,  "Low 
Temperatures  in  the  Isothermal  Region."  We  infer  that  in 
the  general  circulation  there  are  heat  maxima,  or  warm  crests, 
which  form  near  the  equator,  Case  I,  and  move  toward  the  pole 
to  middle  latitudes,  Case  II.  The  temperature  gradients  in  the 
isothermal  region  are,  therefore,  very  unsteady  at  any  place,  and 
there  is  a  continuous  mixture  of  the  air  currents,  along  with  a 
vigorous  radiation  of  heat  from  below,  as  will  be  further  indicated. 
The  computed  data  of  Case  II  are  here  produced  in  Tables  29-42. 
While  there  are  instances  of  "antitrade"  winds  at  high  elevations, 
the  "trade"  winds  apparently  penetrate  to  very  high  altitudes 
at  other  times.  These  observations  were  all  made  on  the  eastern 
side  of  the  north  Atlantic  Tropic  Ocean,  and  on  the  eastern  edge 
of  the  high-pressure  section  of  that  region,  where  westward  winds 
from  N.E.-  to  S.W.-ward  prevail  as  part  of  the  forced  circula- 
tion. On  the  western  side  of  the  ocean  and  western  edge  of  this 
same  section,  it  is  probable  that  the  "  antitrade"  wind  will  pre- 
dominate much  more  vigorously,  but  this  is  again  a  localized 
effect  of  temperature  and  pressure  distribution.  An  inspection 
of  the  Victoria  Nyanza  temperatures,  Table  27,  indicate  a 
very  pronounced  fall  of  temperature  on  the  levels  from  16,000 
to  18,000  meters,  such  as  would  produce  a  violent  westward 
circulation,  which  does  not  seem  to  exist.  It  would  be  proper 
to  confirm  these  valuable  observations  at  other  points  over  the 
equator,  whenever  practical.  Similarly  the  temperatures  of 
St.  Louis  relative  to  Europe  would  demand  a  violent  westward 
wind  at  the  high  levels,  which  likewise  does  not  regularly  exist. 
These  facts  show  how  difficult  it  is  to  construct  a  satisfactory 
system  of  temperatures  and  velocities  for  the  planetary  circula- 
tion. There  is  great  need  for  high-level  balloon  ascensions 
recording  temperature,  humidity,  and  wind  velocity  and  direc- 
tion, upon  which  to  base  the  computations  for  the  other  terms 
in  the  problems  of  the  atmosphere. 


118  THERMODYNAMIC  METEOROLOGY 

The   Thermodynamic   Tables   of  the  Planetary   Circulation   and 

Radiation 

After  the  preceding  explanations  regarding  the  observational 
data  and  formulas,  the  reader  can  easily  study  the  results  of 
the  computations  for  Case  II,  in  Tables  29-42,  so  that  only 
special  points  of  interest  will  be  indicated. 

Table  29.  The  pressure  maximum  is  near  latitude  30°  at  all 
elevations;  the  minimum  near  the  pole  is  much  lower  than  that 
at  the  equator. 

Table  30.  The  temperature  maximum  is  in  the  high-press- 
ure maximum  throughout  the  convectional  region,  and  in  the 
isothermal  region  it  moves  from  near  the  equator  toward  the  pole; 
there  is  a  sharp  drop  in  the  temperature  in  passing  from  the 
convectional  to  the  isothermal  region;  this  boundary  is  located 
at  9,000-12,000  meters  on  the  poleward  side  of  40°  and  it  lies 
between  12,000  and  16,000  in  the  tropics;  when  the  temperatures 
in  the  isothermal  region  are  relatively  cold  the  boundary  is 
at  high  elevations,  and  when  warm  at  low  elevations  respectively; 
over  anticyclones  the  isothermal  region  is  at  high  altitudes, 
and  over  cyclones  at  low  latitudes;  it  is  high  in  winter  and  low 
in  summer;  its  elevation  depends  upon  the  temperature  and 
gravitation  conditions  in  the  convection  region  and  not  on  any 
inherent  forces  of  its  own;  it  is  distinguished  in  its  physical 
properties  from  the  convectional  by  some  properties  which  will 
be  indicated  under  the  topic  of  radiation. 

Table  31.  The  density  has  nearly  the  same  value  on 
the  same  level  of  the  tropics  as  a  broad  minimum,  and  it  in- 
creases toward  the  pole,  much  more  in  the  convectional  than  in 
the  isothermal  region. 

Table  32.  The  gas  coefficient  and  the  dependent  specific 
heat  are  variables,  though  assumed  to  be  constant  at  the  surface 
before  radiation  changes  it  with  the  elevation,  but  decreas- 
ing upward  generally,  much  more  near  the  pole  than  over 
the  equator;  there  is  an  irregularity  in  passing  to  the  isothermal 
region,  accompanied  by  the  change  of  temperature  and  velocity 
of  circulation.  The  check  P  =  p  T  R  is  confirmed  at  every 
point. 


THERMODYNAMIC   TABLES 


119 


TABLE  29 

THE    PLANETARY    CIRCULATION    AND    RADIATION 
The  pressure  P  in  the  units  of  force  (M.  K.  S.) 


• 

90° 

80° 

70° 

60° 

50° 

40° 

30° 

20° 

10° 

0° 

19000 

5795 

5854 

5964 

6184 

6356 

6587  j  6945 

6832 

6637 

6445 

18000 

6768 

6838 

6980 

7224 

7409 

7709    8128 

8038 

7848 

7683 

17000 

7922 

8003 

8148 

8433 

8672 

9023  i  9550 

9457 

9273 

9132 

16000 

9247 

9367 

9536 

9843 

10110 

10519 

11177 

11127 

10955 

10829 

15000 

10823 

10935 

11136 

11521 

11833 

12311 

13133 

13084 

12915 

12807 

14000 

12667 

12797 

13028 

13448 

13794 

14408 

15407 

15374 

15215 

15108 

13000 

14825 

14978 

15248 

15740 

16144 

16837 

18041 

18028   17869 

17754 

12000 

17351 

17530 

17846 

18422 

18821 

19674 

21076 

21069   20899 

20782 

11000 

20308 

20513 

20887 

21560 

22028 

22972 

24555 

24523 

24345 

24224 

10000 

23768 

24013 

24405 

25135 

25781 

26808 

28481 

28403 

28220 

28098 

9000 

27818 

28105 

28596 

29367 

30099 

31177 

32860 

32726 

32542 

32418 

8000   32509 

32838 

33367 

34213 

35012 

36068 

37721 

37534 

37344 

37222 

7000  i  37859 

38226 

38772 

39666 

40517 

41530 

43113 

42877 

42678 

42558 

6000 

43930 

44344 

44876 

45781 

46666 

47620 

49111 

48812 

48605 

48495 

5000 

50811 

51251 

51742 

52631 

53516 

54414 

55772 

55427 

55219 

55110 

4000 

58606 

59038 

59464 

60297 

61156 

61967 

63164 

62791 

62576 

62491 

3000 

67437 

67816 

68143 

68867 

69678 

70378 

71355 

70963 

70772 

70707 

2000  i  77432 

77703 

77877 

78455 

79164 

79722 

80428 

80046 

79862 

79838 

1000  !  88730 

88838 

88836 

89168 

89704 

90104 

90462 

90126 

89958 

89950 

000 

101521 

101388 

101135 

101149 

101414 

101588 

101548 

101216 

101042 

101058 

Formula:     log  P  =  log  Po 


nk 


(log  r  -  log  To). 


(182) 


TABLE  30 
THE  TEMPERATURE  T  IN  ABSOLUTE  DEGREES  CENTIGRADE 


z 

90° 

80° 

70° 

60° 

50° 

40° 

30° 

20° 

10° 

0° 

19000 

219.0 

219.2 

219.4 

220  .4 

220.8 

218.6 

214.8 

210.9 

203.0 

193.0 

18000 

218.6 

218.8      219.0 

220.0 

220.6 

218.4 

214.6 

210.6 

204.0 

196.0 

17000      218.3 

218.5      218.7 

219.7 

220.4 

218.2 

214.4 

210.3 

205.0 

199.0 

16000      218.0 

218.2 

218.4 

219.4 

220.2 

218.0 

214.2 

210.0 

206.0 

202.0 

15000 

217.7 

217.9 

218.1 

219.1 

220.0 

217.8 

214.0 

211.0 

208.0 

205.0 

14000 

217.4 

217.6 

217.8 

218.8 

219.8 

218  0 

215.0 

213.0 

210.0 

209.0 

13000 

217.2 

217.4 

217.6 

218.6 

219.6 

219.0 

218.0 

216.2 

215.0 

214.0 

12000 

217.0 

217.2 

217.4 

218.4 

219.4 

220.0 

221.0 

222.0 

221.0 

220.0 

11000 

216.8 

217.0 

217.2 

218.2 

219.2 

221.0 

226.4 

228.3 

227.0 

226.0 

10000 

216.6 

216.8 

217.0 

218.0 

219.0 

222.1 

234.3 

236.3 

235.5 

234.5 

9000 

216.4 

217  0 

218.5 

220.3 

222.2 

230.4 

243.6 

245.4 

244.2 

243.2 

8000 

221.8 

222.1 

224.5 

227.3 

230.0 

238.6 

251.6 

253.2 

252.5 

251.5 

7000      227.0 

227.5 

230.5 

234.8 

238.0 

246.2 

259.2 

260.5 

259.5 

258.5 

6000 

232.8 

233.0 

237.0 

241.8 

245.7 

253.0 

265.8 

266.3 

265.5 

264.8 

5000 

237.3 

239.0 

243.0 

248.3 

253.0 

259.5 

271.7 

271.5 

270.5 

269.5 

4000 

241.5 

244.2 

248.3 

254.2 

259.0 

265.8 

« 

277.3 

276  5 

275.5 

274.0 

3000 

245.3 

248.7 

253.3 

259.7 

264.8 

271.3 

282.9 

281.4 

280.0 

279.0 

2000 

249.0 

253.0 

257.7 

264.6 

270.6 

276.6 

288.4 

286.2 

285.0 

284.0 

1000 

252.5 

257.0 

261.9 

269.0 

276.0 

281.6 

292.0 

290.0 

289.5 

289.0 

000 

255.0 

260.0 

265.0 

273.0 

281.0 

288.0 

299.5 

298.5 

298.7 

298.0 

The  observations  made  in  balloon  and  kite  ascensions. 


120  THERMODYNAMIC  METEOROLOGY 

Table  33.  The  adopted  wind  velocity  indicates  an  east- 
ward movement  with  the  positive  sign  (+),  westward  with 
the  negative  sign  (— ),  and  alternating  with  the  (=F)  signs;  it 
is  quite  likely  that  further  observations  will  enable  us  to  improve 
this  mean  table  of  velocities;  these  must  ultimately  be  so  ad- 
justed to  the  air  masses  associated  with  them  that  the  sum  of  the 
momenta  of  rotation  about  the  earth's  axis  shall  be  equal  to  zero 
in  order  that  the  period  of  the  earth's  rotation  may  be  constant, 
as  indicated  by  astronomical  observations,  and  this  involves 
the  corresponding  pressure,  temperature,  density,  and  radiation 
of  heat  from  point  to  point  throughout  the  entire  atmosphere. 

Table  34.  The  kinetic  energy  of  circulation  from  one 
level  to  another  acts  as  a  balance  in  the  action  of  gravitation 
against  the  pressure  and  heat  terms  in  the  general  equation. 
This  action  is  very  pronounced  in  passing  from  the  convectional 
to  the  isothermal  region;  it  is  strong  in  some  parts  of  cyclones 
and  anticyclones,  and  in  tornadoes,  being  due  to  rapid  changes  in 
the  temperatures  for  short  distances.  When  the  pressure  and 
heat  terms  are  deficient,  the  kinetic  energy  makes  it  up  by  an 
increase  in  the  velocity;  when  in  excess,  it  balances  the  same  by 
decreasing  the  velocity;  the  numerous  horizontal  currents  in 
the  atmosphere,  as  seen  by  the  cloud  motions  in  different  direc- 
tions, exhibit  this  process  which  is  incessantly  at  work  adjusting 
these  delicate  differences  between  pressure  and  radiation  to  the 
controlling  force  of  gravitation ;  these  adjust  themselves  mutually 
at  every  point  in  the  atmosphere,  and  are  not  propagated  at 
long  range  from  one  distant  point  to  another. 

Table  35.  The  hydrostatic  pressure  per  unit  density  is 
computed  from  Tables  29,  31,  for  use  in  the  general  equation 
(196),  which  must  always  be  satisfied.  It  should  be  carefully 
remembered  that  the  density  is  to  be  computed  by  equation 
(176),  and  not  by  equation  (175).  The  term  decreases  upward, 
and  generally  from  the  equator  to  the  pole,  though  there  is 
a  small  maximum  in  the  convectional  region  just  north  of  the 
high-pressure  belt,  in  the  levels  5,000  to  9,000  meters. 

Table  36.  The  free  heat  on  which  radiation  depends  gives 
the  change  in  the  heat  contents  of  the  unit  mass  per  1,000 


THERMODYNAMIC   TABLES 


121 


TABLE  31 

THE  PLANETARY  CIRCULATION  AND  RADIATION 
The  density  p  in  kilograms  per  cubic  meter 


z 

90° 

80° 

70° 

60° 

50° 

40° 

30° 

20° 

10° 

O8 

19000 

0.1810 

0.1788 

0.1776 

0.1769 

0.1754 

0.1756 

0.1753 

0.1737 

0.1700 

0.1670 

18000 

.2022 

.1997 

.1986 

.1976 

.1956 

.1964 

.1961 

.1950 

.1915 

.1892 

17000 

.2261 

.2233 

.2217 

.2206 

.2188 

.2197 

.2199 

.2189 

.2156 

.2140 

16000 

.2524 

.2498 

.2480 

.2463 

.2440 

.2456 

.2459 

.2458 

.2428 

.2416 

15000 

.2823 

.2788 

.2768 

.2754 

.2729 

.2740 

.2758 

.2758 

.2730 

.2722 

14000 

0.3157 

0.3118 

0.3096 

0.3074 

0.3043 

0.3064 

0.3090 

0.3093 

0.3066 

0.3061 

13000 

.3531 

.3487 

.3463 

.3437 

.3403 

.3423 

.3457 

.3464 

.3438 

.3433 

12000 

.3949 

.3900 

.3872 

.3835 

.3796 

.3824 

.3861 

.3870 

.3843 

.3840 

11000 

.4417 

.4362 

.4331 

.4300 

.4245 

.4270 

.4304 

.4311 

.4284 

.4282 

10000 

.4939 

.4878 

.4844 

.4796 

.4748 

.4765 

.4783 

.4786 

.4758 

.4759 

9000 

0.5524 

0.5456 

0.5415 

0.5357 

0.5300 

0.5306 

0.5295 

0.5293 

0  .  5265 

0.5269 

8000 

.6172 

.6094 

.6043 

.5972 

.5902 

.5885 

.5841 

.5835 

.5806 

.5813 

7000 

.6878 

.6790 

.6724 

.6634 

.6548 

.6505 

.6424 

.6414 

.6385 

.6394 

6000 

.7645 

.7546 

.7660 

.7346 

.7240 

.7170 

.7047 

.7033 

.7003 

.7016 

5000 

.8479 

.8364 

.8255 

.8112 

.7981 

.7884 

.7714 

.7698 

.7669 

.7684 

4000 

0.9384 

0.9249 

0.9114 

0.8935 

0.8775 

0.8647 

0  .  8428 

0.8413 

0.8382 

0.8402 

3000 

1.0370 

1.0207 

1.0041 

0.9821 

0.9628 

0.9466 

0.9191 

0.9177 

0.9148 

0.9173 

2000 

1  .  1440 

1  .  1244 

1.1071 

1  .  0775 

1.0543 

1  .  0344 

1  .  0008 

0.9998 

0.9969 

1  .  0001 

1000 

1.2604 

1.2367 

1.2125 

1.1802 

1  .  1523 

1.1284 

1  .  0880 

1.0878 

1.0850 

1.0886 

000 

1.3870 

1.3586 

1.3296 

1.2908 

1.2574 

1.2289 

1.1812 

1.1814 

1.1785 

1  1825 

Formula:     log  p 


log  Po  +  £—  j-  (log  T  -  log  To). 


(183) 


TABLE   32 
THE  GAS  COEFFICIENT  R,  WHICH  Is  A  VARIABLE 


z 

90° 

80° 

70° 

60° 

50° 

40° 

30° 

20° 

10° 

0° 

19000 

146.14 

149.37 

153.03 

158.60 

164.12 

171.56 

184.39 

186.48 

192  .  33 

199.92 

18000 

153  .  12 

156.51 

160.45 

166.18 

171.70 

179.71 

193.14 

195.72 

200.88 

207.12 

17000 

160.47 

164.01 

168.01 

174.01 

179  .  86 

188.24 

202  .  54 

205.43 

209.77 

214.42 

16000 

168.03 

171.88 

176.06 

182.21 

188.17 

196.94 

217  .  10 

215.62 

219.05 

221.92 

15000 

176.09 

179.98 

184.36 

190.95 

197.10 

206.29 

222.48 

224  .  88 

227.53 

229.53 

14000 

184.53 

188.61 

193.20 

199.95 

206.21 

215.68 

231.91 

233.39 

236.28 

236.14 

13000 

193.29 

197.56 

202.37 

209.44 

216.00 

224.58 

239.38 

240.73 

241.75 

241.63 

12000 

202.46 

206.94 

211.93 

219.37 

225.99 

233.85 

246.98 

245.24 

246  .  08 

245.98 

11000 

212.07 

216.76 

222  .  04 

229.79 

236.72 

243.45 

251.97 

249.17 

250.38 

250.28 

10000 

222.14 

227.05 

232.58 

246.01 

247.95 

253.29 

254.13 

250.64 

251.86 

251.77 

9000 

232  .  68 

237.39 

241.68 

248.85 

255.56 

255.06 

254.74 

251.96 

253  .  09 

253.01 

8000 

237.48 

242.61 

245.94 

252  .  06 

257.92 

256.88 

256.66 

254.07 

254.69 

254.62 

7000 

242.48 

247.47 

250.16 

254.65 

259  .  99 

259.30 

258.96 

256.62 

257.57 

257.51 

6900 

247.35 

252.22 

253.79 

257.74 

262.33 

262.51 

262.21 

260  .  62 

261.39 

261.05 

5000 

252  .  54 

256.39 

257.92 

261.31 

265.05 

265.98 

266.11 

265.19 

266.19 

266.15 

4000 

258.59 

261.39 

262.76 

265.47 

269.08 

269  .  62 

270.28 

269.94 

270.97 

271.46 

3000 

265.12 

267  .  15 

267.91 

270.02 

273.29 

274.04 

274.43 

274.72 

276.27 

276.28 

2000 

271.82 

273.15 

273.71 

275.18 

277.48 

278.65 

278.66 

279.74 

281.07 

281.11 

1000 

278.81 

279  .  51 

279.75 

280.88 

282.05 

283.56 

284.74 

285.70 

286.38 

285.93 

000 

287.03 

287.03 

287  .  03 

287.03 

287  .  03 

287  .  03 

287.03 

287.03 

287.03 

287.03 

Formula:     log  R  =  log  Ro  +  (n-l)  (log  T  -  log  To).     .     .     .     (184) 
Check     P  =   Tp  Rat  every  point (173) 


122  TIIERMODYNAMIC   METEOROLOGY 

meters,  and  the  table  indicates  how  far  the  atmosphere  has  de- 
parted from  the  adiabatic  state.  The  term  increases  upward, 
and  generally  from  the  equator  to  the  pole,  but  there  is  a  mini- 
mum on  entering  the  isothermal  region  in  middle  latitudes  and 
a  region  of  marked  irregular  progression  in  the  values.  Un- 
fortunately there  is  no  way  to  compute  this  term  directly,  as  it 
depends  upon  the  evaluation  of  the  specific  heat  and  the  velocity 
of  the  circulation,  by  equation  (199).  It  is  therefore  a  great  loss 
to  science  when  an  observatory  measures  the  temperatures,  but 
not  the  humidity  and  wind  velocity,  at  different  levels,  because 
the  entire  subject  of  thermodynamic  meteorology  is  thereby 
excluded  from  further  discussions.  The  magnitude  of  the  term 
(Qi  —  Qo)  can  be  seen  in  Table  36  to  be  very  great  in  the  upper 
levels,  and  that  in  comparison  with  it  the  kinetic  energy  of  the 
circulation,  as  in  Table  34,  is  very  small.  Hence,  all  those  the- 
ories of  atmospheric  circulation  which  depend  upon  gravitation, 
pressure,  and  circulation  alone,  and  omit  heat  changes  through 
radiation,  either  absorption  or  emission,  have  no  permanent 
value.  The  difficulty  of  determining  the  heat  term  has,  no  doubt, 
been  the  cause  of  this  defect  which  prevails  in  meteorological 
literature,  but  it  is  none  the  less  a  fatal  defect  in  this  branch  of 
science.  Table  43,  p.  131,  gives  the  second  differences  of  the 
heat  contents,  and  it  is  the  rate  of  change  of  the  heat  per  1,000 
meters,  or  the  radiation  rate,  which  prevails  on  the  average  in 
all  parts  of  the  atmosphere,  and  it  is  fundamental  to  all  studies 
of  "solar  constant "  and  solar  insolation.  This  table  will  be  dis- 
cussed more  fully  under  the  subject  of  Bolometry  and  Pyrhe- 
liometry,  and  at  this  place  only  its  leading  features  will  be  noted. 
Taking  the  hemisphere  as  a  whole,  the  mean  rate  of  radiation  is 
—  157.1  in  the  convectional  region,  and  —  283.1  in  the  isother- 
mal region,  almost  twice  as  great  in  the  latter.  This  changes 
greatly  in  latitude,  as  can  be  seen  in  the  means  for  the  isothermal 
region  (/),  and  the  convectional  region  (C),  respectively.  The 
means  on  the  table  are  taken  outside  the  layers  of  transition,  and 
these  give  nearly  a  constant  value  in  C  from  the  equator  to  lati- 
tude 60°,  but  an  increasing  value  to  the  pole;  in  the  /  region  there 
is  a  minimum  in  the  tropics,  maximum  in  the  middle  latitudes, 


THERMODYNAMIC   TABLES 


123 


TABLE  33 

THE  PLANETARY  CIRCULATION  AND  RADIATION 
The  velocity  q  in  meters  per  second 


z 

90° 

80° 

70° 

60° 

50° 

40° 

30° 

20° 

10° 

0° 

19000 

0.0 

3.0 

4.0 

4.0 

5.0 

T5.0 

T6.0 

T7.0 

+  9.0 

-12.0 

18000 

0.0 

3.0 

4.0 

5.0 

6.0 

6.0 

T7.0 

=F8.0 

+~10.0 

-12.0 

17000    0.0 

3.0 

4.0 

6.0 

7.0 

7.0 

T8.0 

+9.0 

+10.0 

-12.0 

16000 

0.0 

4.0 

5.0 

7.0 

8.0 

9.0 

T6.0 

T6.0 

T10.0 

-11.0 

15000 

0.0 

5.0 

6.0 

8.0 

9.0 

10.0 

T4.0 

+4.0 

T  7.0 

-10.0 

14000 

0.0 

6.0 

7.0 

9.0 

10.0 

11.0 

T2.0 

T2.0 

=F  4.0 

-  9.0 

13000 

0.0 

6.0 

8.0 

10.0 

11.0 

12.0 

4.0 

5.0 

T  2.0 

-  8.0 

12000 

0.0 

7.0 

9.0 

11.0 

12.0 

13.0 

9.0 

5.0 

-  3.0 

-  7.0 

11000 

0.0 

7.0 

11.0 

13.0 

15.0 

16.0 

12.0 

6.0 

-  4.0 

-  5.0 

10000 

0.0 

8.0 

15.0 

17.0 

20.0 

20.0 

15.0 

6.0 

-  5.0 

-  3.0 

9000 

0.0 

5.6 

25.2 

31.6 

35.0 

34.0 

21.7 

0.8 

-  3.8 

-  2.0 

8000 

0.0 

4.8 

22.8 

28.9 

31.5 

29.0 

16.0 

.-2.6 

-  5.0 

-  2.5 

7000 

0.0 

4.0 

19.6 

25.8 

26.9 

22.0 

9.0 

-4.9 

-  6.2 

-  2.6 

6000 

0.0 

3.2 

15.9 

21.4 

21.8 

15.4 

2.4 

-6.8 

-  7.0 

-  2.7 

5000 

0.0 

2.0 

12.4 

17.6 

17.2 

11.4 

2.6 

-8.6 

-  7.4 

-  2.8 

4000 

0.0 

0.8 

9.0 

13.8 

13.6 

9.2 

6.6 

-9.4 

-  7.6 

-  2.8 

3000 

0.0 

0.4 

6.6 

11.0 

11.0 

7.4 

7.3 

-9.4 

-  7.2 

-  2.8 

2000 

0.0 

1.6 

5.0 

9.4 

9.4 

6.4 

6.1 

-8.4 

-  6.2 

-  2.4 

1000 

0.0 

2.4 

3.8 

8.0 

8.0 

5.6 

4.1 

-6.7 

-  4.6 

-  1.4 

000 

0.0 

3.4 

3.4 

7.4 

7.1 

5.1 

2.0 

-4.6 

-  2.6 

-  0.4 

From  the  observations  made  in  balloon  and  kite  ascensions. 

TABLE  34 
THE  KINETIC  ENERGY  OF  CIRCULATION  —  X  (22i  —  <Z2o) 


z 

90° 

80° 

70° 

60° 

50° 

40° 

30° 

20° 

10° 

0° 

19000 

0.0 

0.0 

0.0 

+     4.5 

+     5.5 

+     5.5 

+     6.5 

+  7.5 

+  9.5 

0.0 

18000 

0.0 

0.0 

0.0 

+     5.5 

+     6.5 

+     6.5 

+     7.5 

+  8.5 

0.0 

0.0 

17000 

0.0 

+  3.5 

+  8.5 

+     6.5 

+     7.5 

+  16.0 

-  14.0 

-22.5 

0.0 

-11.5 

16000 

0.0 

+  4.5 

+  5.5 

+     7.5 

+     8.5 

+     9.5 

-  10.0 

-10.0 

-25.5 

-10.5 

15000 

0.0 

+  5.5 

+  6.5 

+     8.5 

+     9.5 

+  10.5 

-     6.0 

-  6.0 

-16.5 

-  9.5 

14000 

0.0 

0.0 

+  7.5 

+     9.5 

+  22.0 

+  11.5 

+     6.0 

+10.5 

-  6.0 

-  8.5 

13000 

0.0 

+  6.5 

+  8.5 

+  10.5 

+  11.5 

+  12.5 

+  32.5 

+  5.0 

+  2.5 

-  7.1 

12000 

0.0 

0.0 

+20.0 

+  24.0 

+  40.5 

+  43.5 

+  31.5 

+  5.5 

+  3.5 

-12.5 

1  1000 

0.0 

+  7.5 

+52.0 

+  60.0 

+  87.5 

+  72.0 

+  40.5 

-  0.0 

+  4.5 

-  8.0 

10000 

0.0 

+16.3 

-54.9 

+354.8 

+412.5 

+378.0 

+123.0 

-17.7 

-  5.3 

-  2.5 

9000 

0.0 

-  4.2 

-57.6 

-81.3 

-116.4 

-157.5 

-107.4 

+  3.1 

+  5.3 

+  1.1 

8000 

0.0 

-  3.5 

-67.8 

-  84.8 

-134.3 

-178.5 

-  87.5 

+  8.6 

+  6.7 

+  0.3 

7000 

0.0 

-  2.9 

-65.7 

-103.8 

-124.2 

-123.4 

-  37.6 

+11.1 

+  5.3 

+  0.2 

6000 

0.0 

-  3.1 

-49.5 

-  74.1 

-  89.7 

-  53.6 

+     0.5 

+13.9 

+  2.9 

+  0.2 

5000 

0.0 

-  1.7 

-36.4 

-  59.7 

-  55.4 

-  22.7 

+  18.4 

+  7.2 

+  1.5 

0.0 

4000 

0.0 

-  0.2 

-18.7 

-  34.7 

-  32.0 

-14.9 

+     4.8 

0.0 

-  3.0 

0.0 

3000 

0.0 

+  1.2 

-  9.3 

-  16.3 

-  16.3 

-     6.9 

-     8.0 

-  8.9 

-  6.7 

-  1.0 

2000 

0.0 

+  1.6 

-   5.3 

-  12.2 

-  12.2 

-     4.8 

-  10.2 

-12.8 

-  8.6 

-  1.9 

1000 

0.0 

+  2.9 

-   1.4 

-     4.6 

-     6.8 

-     2.7 

-     6.4 

-11.8 

-  7.2 

-  0.9 

000 

0.0 

To  obtain  the  kinetic  energy  these  numbers  must  be  multiplied  by  the  appropriate  values 
of  p:o. 


124  THERMODYNAMIC   METEOROLOGY 

and  minimum  in  the  polar  zone.  The  broad  fact  is  clear  that 
about  twice  as  much  heat  radiates  through  the  isothermal  region 
as  through  the  convection  region.  At  the  strata  of  transition  the 
radiation  is  in  some  latitudes  positive,  as  at  the  10,000-meter 
level  from  30°  to  60°  latitude.  These  special  sources  of  heat 
will  be  further  considered  in  discussing  the  value  of  the  "solar- 
constant"  of  radiation.  The  importance  of  such  data  in  radia- 
tion problems  is  very  great,  because  it  seems  to  explain  several 
processes  in  the  atmosphere  that  have  been  very  obscure. 

Table  37.  The  entropy  increases  upward  and  from  the 
equator  to  the  pole,  though  the  same  variation  occurs  in  the 
transition  to  the  isothermal  region  as  was  noted  for  the  heat 
contents. 

Table  38.  The  work  against  external  forces ,  such  as  expansion 
when  a  mass  of  air  is  raised  from  one  level  to  another  in  the 
circulation,  is  to  be  computed  through  a  new  value  of  R',  which 
differs  from  the  R  of  (184)  in  consequence  of  the  elimination 
of  the  velocity  of  the  circulation,  as  explained  in  (199),  (333), 
on  Table  19.  It  is  the  circulation  term  which  continu- 
ally interferes  with  a  perfect  check  to  this  series  of  equations, 
and  on  that  account,  therefore,  must  be  most  carefully 
observed. 

Table  39.  The  inner  energy  is  the  heat  contents  remaining 
after  the  work  has  been  done  externally,  and  it  increases  upward 
and  from  the  equator  to  the  pole,  with  the  same  variation  be- 
tween the  convectional  and  the  isothermal  regions. 

Table  40.  The  radiation  function  is  the  rate  of  change  of  the 
inner  energy  with  the  change  of  the  volume,  and  it  may  be 
computed  by  several  formulas.  The  K\Q  differs  from  Pi0  by  the 
rate  of  change  of  the  heat  contents  with  the  change  in  volume, 
or  it  may  be  seen  to  depend  upon  the  variation  of  the  gas  coeffi- 
cient R.  The  value  of  Ki0  generally  decreases  with  the  altitude, 
and  there  is  a  small  maximum  in  the  high-pressure  belt  of  the 
convectional  region,  but  this  shifts  toward  the  pole  in  the  iso- 
thermal region.  The  immediate  problem  is  to  determine  the 
relation  to  the  temperature  through  the  approximate  Stefan 
formula,  and  this  has  been  done  in  the  same  way  as  that  followed 


THERMODYNAMIC    TABLES 


125 


TABLE  35 

THE  PLANETARY  CIRCULATION  AND  RADIATION 

Pi  - 


The  Hydrostatic  Pressure  per  Unit  Density  — 


PlO 


z 

90° 

80° 

70° 

60° 

50° 

40° 

30° 

20° 

10° 

0° 

19000 

5078.2 

5198.1 

5401  .  4 

5552.5 

5676.6 

6032.3 

6370.4 

6540.2 

6697.9 

6651  .  2 

18000 

5387.5 

5508.6 

5556.6 

5782.0 

6095.6 

6314.3 

6836.6 

6855.0 

6998.9 

7187.5 

17000 

5537.1 

5765.0 

5909.0 

6038.5 

6214.4 

6428.8 

6985.8 

7185.8 

7338.7 

7449.5 

16000 

5893  .  9 

5932.7 

6082.3 

6431.6 

6665.4 

6897.7 

7497.2 

7503  .  8 

7600.0 

7699.7 

15000 

6167.3 

6305.5 

6466.7 

6532.2 

6794.8 

7226.0 

7777.0 

7826.5 

7936.5 

7956.5 

14000 

6453  .  3 

6603  .  1 

6766.3 

6962.3 

7291.3 

7490  .  0 

8045.2 

8094.0 

8161.2 

8149.2 

13000 

6754.0 

6908.4 

7082  .  8 

7366.2 

7436.2 

7828.3 

8294  .  7 

8293.0 

8321.8 

8325.7 

12000 

7069.0 

7220.8 

7413.5 

7704.4 

7975.7 

8149.3 

8520.6 

8445.0 

8479.5 

8475.8 

11000 

7396.3 

7575.8 

7668.0 

7860.7 

8345.6 

8490.4 

8640  .  0 

8531.2 

8571.2 

8569.0 

10000 

7740.8 

7919.5 

8169.8 

8335.6 

8594.6 

8675.4 

8690  .  2 

8577.4 

8623.2 

8615.8 

9000 

8021.6 

8195.8 

8327.8 

8555.8 

8771.6 

8740.2 

8730.4 

8641.2 

8674.0 

8669.8 

8000 

8199.2 

8364.0 

8466.6 

8643  .  6 

8843.4 

8816.8 

8799.4 

8724  .  6 

8750.0 

8742.0 

7000 

8359.8 

8595.0 

8606.8 

8748.2 

8919.2 

8906.2 

8897.0 

8826.6 

8854  .  0 

8854  .  6 

6000 

8535.2 

8702.6 

8737.6 

8862.8 

9001.4 

9026.2 

9025.6 

8980.5 

9015.7 

9000  .  0 

6000 

8727.2 

8842  .  8 

8892.2 

8993  .  6 

9119.0 

9137.4 

9158.6 

9141.0 

9166.4 

9177.0 

4000 

8940  .  8 

9023.6 

9061.5 

9138.4 

9261  .  0 

9287.8 

9297.4 

9291.6 

9350.8 

9349.0 

3000 

9165.4 

9217.6 

9234.4 

9310.4 

9405.0 

9433.6 

9451.0 

9473.2 

9509.4 

9524.5 

2000 

9397.8 

9432  .  0 

9461.2 

9490  .  8 

9553.0 

9600.4 

9607.2 

9657.0 

9698.4 

9682.0 

1000 

9663.0 

9671.6 

9676.5 

9697.5 

9719.4 

9743.6 

9770  .  8 

9776.8 

9793.3 

9781.4 

000 

.... 

Computed  from  Tables  29,  31. 

TABLE  36 
THE  HEAT  WHICH  GIVES  RISE  TO  RADIATION  —  (Q\  —  Q0) 


z 

90° 

80° 

70° 

60° 

50° 

40° 

30° 

20° 

10° 

0° 

19000 

4694.2 

4581.1 

4451.2 

4253.8 

4064.2 

3800.2 

3350  .  6 

3270.0 

3079.6 

2853.1 

18000 

4449.3 

4331.0 

4195.5 

3982.7 

3794.3 

3514.3 

2990.1 

2944.8 

2791.4 

2605.0 

17000 

4204.4 

4064.9 

3920.2 

3707.8 

3512.0 

3210.4 

2736.1 

2677  .4 

2481.0 

2363.8 

16000 

3927.8 

3793.2 

3643.8 

3424  .  3 

3216.4 

2908.5 

2391.6 

2291.5 

2203.3 

2104.8 

15000 

3645.9 

3504.4 

3350.2 

3140.1 

2907.2 

2587.5 

2050.2 

1934.7 

1900.1 

1861.0 

14000 

3352.2 

3131.0 

3041.4 

2803.6 

2571.7 

2273.9 

1749.7 

1696.7 

1646.3 

1653.7 

13000 

3045.8 

2811.2 

2719.8 

2470.7 

2244  .  6 

1962.7 

1465.6 

1499.7 

1470.5 

1483.8 

12000 

2724.1 

2568.5 

2372.2 

2109.6 

1866.1 

1609.4 

1233.1 

1355.2 

1322.5 

1341.5 

11000 

2388.8 

2217.5 

1988.4 

1714.0 

1439.6 

1248.7 

1120.4 

1268.5 

1222.6 

1238.1 

10000 

2036.9 

1888.9 

1759.7 

1093.8 

792.7 

744.4 

990.6 

1238.5 

1185.9 

1185.9 

9000 

1774.7 

1611.1 

1534.0 

1331.4 

1151.3 

1218.7 

1177.8 

1159.1 

1126.8 

1133.6 

8000 

1607.5 

1438.2 

1399.5 

1235.4 

1093.6 

1167.3 

1085.8 

1074.0 

1049.0 

1057.6 

7000 

1439.1 

1273.3 

1263.3 

1157.2 

1008.1 

1016.1 

941.2 

959.0 

936.0 

948.0 

6000 

1267.2 

1121.2 

1114.6 

1013.9 

887.3 

832.2 

781.0 

810.4 

791.2 

800.4 

5000 

1075.2 

958.1 

948.3 

867.6 

736.2 

697.8 

625.3 

658.0 

628.9 

622.9 

4000 

860.2 

777.7 

760.0 

693.8 

573.4 

534.2 

496.8 

501.1 

461.1 

449.7 

3000 

634.2 

575.6 

563.7 

559.4 

414.2 

372.3 

366.3 

342.5 

292.4 

285.9 

2000 

400.3 

364.2 

357.3 

319.7 

260.5 

202  .4 

192.4 

160.1 

121.7 

122.1 

1000 

140.4 

125.2 

125.8 

109.7 

91.9 

62.1 

45.5 

34.6 

18.5 

24.4 

000 

Formula:      -  (Qi  -  Qo)  =  -  (Cp*  - 
Check     q  (zi  -  Zo)  = l- 


(T*    -  To)  +  \  (<Zi2  -  <Z02) 
-  Q0) 


(199). 

(196). 


Comparing  the  numerical  values  of  these  terms,  it  is  seen  how  impossible  it  is  to  solve 
any  problems  in  dynamic  meteorology  without  the  term  —  (Qi  —  Qo). 


126  THERMODYNAMIC  METEOROLOGY 

on  Table  20,  the  intermediate  steps  of  computing  log  C  and  A 
for  the  ratios  Ki/KQ  and  Ti/To  being  here  omitted. 

Table  41.  The  log  c  in  the  formula  diminishes  slowly  from  the 
surface  upward,  it  increases  from  the  equator  to  the  pole  in 
the  convectional  region,  but  decreases  in  the  isothermal  region, 
the  result  being  a  nearly  uniform  value  on  the  high  levels  of  the 
isothermal  region.  The  minus  sign  affects  only  the  characteristic 
of  the  logarithm,  and  for  a  mean  value  log  c  =  —  5.500,  c  = 
3.162X  10~5.  The  Kurlbaum  coefficient  in  the  Stefan  formula 
for  a  perfect  radiator  is  taken  at  7.68  X  10~n  (C.  G.  min.  C°) 
=  5.32  X  10~6  joules  per  square  meter  per  sec.,  so  that  the 
air  radiates  at  six  times  the  rate  of  a  perfect  radiator  in  the 
ether,  as  when  the  sun  transmits  its  radiant  energy  to  the  earth. 
The  quantity  c  relates  to  a  summation  of  the  successive  radia- 
tions from  one  air  volume  to  another  throughout  a  layer  1,000 
m.  deep.  The  number  of  repeated  transfers  in  this  depth  can 
only  be  approximately  inferred  from  the  knowledge  that  the 
efficient  radiating  layer  of  the  atmosphere,  or  that  for  which 
any  further  increase  of  depth  does  not  add  to  the  radiant  in- 
tensity, amounts  to  several  meters.  The  latter  quantity  is  con- 
siderably smaller  than  the  superficial  radiation  from  a  black 
solid. 

Table  42.  The  exponent  a  diminishes  from  the  surface  up- 
ward; it  increases  from  the  equator  to  the  pole  in  the  convec- 
tional region,  but  diminishes  in  the  isothermal  region.  Its 
mean  value  is  3.82  instead  of  4.00,  as  in  the  Stefan  formula, 
at  the  surface.  Comparing  the  data  we  have  for  the 

Perfect  Radiator,  Kw  =5.32  X  10~6  TV'00    joule/sq.  m.  sec. 

(K.  M.  S.  system)  and  for  the 
Atmosphere,  K10  =  3.162  X  10~5  Tic3'82  per  1,000  m. 

The  consequences  of  these  data  in  theories  of  atmospheric 
radiation  will  require  much  careful  investigation  in  several 
directions. 


THERMODYNAMIC   TABLES 


127 


TABLE  37 

THE  PLANETARY  CIRCULATION  AND  RADIATION 
The  Entropy  (5,  -  S0) 


2 

90° 

80° 

70° 

60° 

50° 

40° 

30° 

20° 

10° 

0° 

19000 

-21.454 

-20.919 

-20.307 

-19.318 

-18.415 

-17.392 

-15.606 

-15.516 

-15.133 

-14.669 

18000 

-20.363 

-19.799 

-19.166 

-18.111 

-17.208 

-16.099 

-13.938 

-13.993 

-13.650 

-13.190 

17000 

-19.269 

-18.612 

-17.933 

-16.885 

-15.942 

-14.720 

-12.768 

-12.740 

-12.073 

-11.790 

16000 

-18.026 

-17.392 

-16.692 

-15.615 

-14.613 

-13.348 

-11.171 

-10.884 

-10.644 

-10.343 

15000 

-16.755 

-16.090 

-15.363 

-14.338 

-13.221 

-11.875 

-  9.558 

-  9.126 

-  9.091 

-  8.990 

14000 

-15.427 

-13.936 

-13.971 

-12.819 

-11.705 

-10.407 

-  8.082 

-  7.906 

-  7.748 

-  7.819 

13000 

-14.029 

-12.937 

-12.505 

-11.308 

-10.226 

-  8.942 

-  6.677 

-  6.845 

-  6.745 

-  6.838 

12000 

-12.559 

-11.831 

-10.917 

-  9.664 

-  8.509 

-  7.299 

-  5.512 

-  6.019 

-  5.904 

-  6.019 

11000 

-11.024 

-10.224 

-  9.159 

-  7.859 

-  6.570 

-  5.635 

-  4.864 

-  5.330 

-  5.287 

-  5.377 

10000 

-  9.408 

—  8.510 

-  8.079 

-  4.990 

-  3.593 

-  3.291 

-  4.146 

-  5.031 

-  4.944 

-  4.965 

9000 

-  8.100 

-  7.337 

-  6.926 

-  5.949 

-  5.092 

-  5.197 

-  4.757 

-  4.649 

-  4.537 

-  4.583 

8000 

-  7.164 

-  6.398 

-  6.152 

-  5.348 

-  4.672 

-  4.816 

-  4.251 

-  4.182 

-  4.098 

-  4.147 

7000 

-  6.268 

-  5.531 

-  5.403 

-  4.836 

-  4.169 

-  4.071 

-  3.728 

-  3.643 

-  3.566 

-  3.623 

6000 

-  5.397 

-  4.751 

-  4.644 

-  4.138 

-  3.558 

-  3.248 

-  2.905 

-  3.014 

-  2.952 

-  2.996 

5000 

-  4.491 

-  3.966 

-  3.861 

-  3.454 

-  2.876 

-  2.589 

-  2.278 

-  2.402 

-  2.304 

-  2.992 

4000 

-  3.534 

-  3.156 

-  3.030 

-  2.700 

-  2.181 

-  1.989 

-  1.774 

-  1.796 

-  1.660 

-  1.626 

3000 

-  2.566 

-  2.295 

-  2.206 

-  2.134 

-  1.547 

-  1.359 

-  1.283 

-  1.207 

-  1.035 

-  1.016 

2000 

-  1.596 

-  1.428 

-  1.375 

-  1.198 

-  0.953 

-  0.725 

-  0.663 

-  0.609 

-  0.424 

-  0.426 

1000 

-  0.553 

-  0.484 

-  0.478 

-  0.405 

-  0.330 

-  0.218 

-  0.154 

-  0.118 

-  0.063 

-  0.083 

000 

Formula  :  (Si  -So) 


-    .     -     (331)  . 


TABLE  38    THE  EXTERNAL  WORK  (Wi  -  W0) 


z 

90° 

80° 

70° 

60° 

50° 

40° 

30° 

20° 

10° 

0° 

19000 

3601.4 

3688.7 

3854.4 

3948  .  5 

4017.9 

4256.9 

4344.3 

4652.0 

4754.7 

4942.6 

18000 

3849.6 

3926.9 

3935.8 

4099.8 

4358  .  9 

4496.7 

4867.6 

4872.0 

4972  .  4 

5107.2 

17000 

3918.8 

4109.2 

4208.6 

4276.8 

4396.1 

4523.4 

4943.4 

5126.4 

5222.6 

5299.5 

16000 

4195.8 

4195.7 

4302.1 

4588.0 

4761.7 

4905.1 

5355.2 

5333  .  0 

5403  .  6 

5474.9 

15000 

4387.8 

4485.0 

4601.7 

4606.5 

4801.8 

5140.7 

5536.4 

5552.7 

5652.6 

5661.2 

14000 

4588.9 

4645.9 

4812.1 

4939.4 

5201.4 

5314.0 

5717.8 

5751.3 

5804  .  0 

5794.1 

13000 

4801.0 

4887.7 

5035.7 

5247.1 

5251.8 

5562.5 

5885.3 

5893.3 

5913.8 

5921.5 

12000 

5023.1 

5129.9 

5266.0 

5581.0 

5681.9 

5781.4 

6044.0 

6003.6 

6028.7 

6030.5 

11000 

5253  .  5 

5383  .  6 

5409.6 

5523  .  0 

5928.7 

6018.3 

6130.8 

6065.7 

6091.6 

6093.9 

10000 

5496.4 

6532.3 

5845.3 

5818.7 

6019.3 

6056.5 

6143.5 

6102.3 

6133.0 

6125.6 

9000 

5701.4 

5828.4 

5938.1 

6107.6 

6271.4 

6259.4 

6237.8 

6143.2 

6166.6 

6164.4 

«000 

5830.8 

5945.7 

6038.0 

6167.7 

6326.5 

6321.2 

6280.2 

6202.0 

6220.2 

6214.7 

7000 

5942.7 

6130.0 

6138.9 

6249.8 

6377.6 

6366.8 

6336.1 

6271.0 

6291.2 

6295.6 

6000 

6068.4 

6193.7 

6226.8 

6322.9 

6424.9 

6433.7 

6418.4 

6381.8 

6411.4 

6398.4 

5000 

6205.0 

6286.7 

6333.3 

6411.4 

6498.8 

6501.0 

6506.4 

6498.2 

6515.2 

6524.1 

4000 

6356.4 

6415.4 

6448.2 

6506.0 

6593.8 

6609.2 

6608.1 

6603.5 

6651.2 

6646.0 

3000 

6515.8 

6551.0 

6564.4 

6639  .  2 

6722  .  8 

6708  .  4 

6724.1 

6739.3 

6761.0 

6774.3 

2000 

6680.6 

6704  .  3 

6731  .  6 

6750.3 

6795.4 

6826  .  1 

6830.0 

6874  .  9 

6900.7 

6884.4 

1000 

6870.7 

6874  .  9 

6880.0 

6896.4 

6911.3 

6928.7 

6951.1 

6953.9 

6965.8 

6955.6 

000 



Formula:     (Wi  -  Wo)  =  R'IO  (T*  -  To)  -      Plo      ....     (333). 

Cp'io    .    .    .      (349)  and  (350). 


128 


THERMODYNAMIC   METEOROLOGY 


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THERMODYNAMIC   TABLES 


129 


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130 


THERMODYNAMIC  METEOROLOGY 


TABLE  41 

THE  PLANETARY  CIRCULATION  AND  RADIATION 
The  coefficient  log  c  in  log  K10  =  log  c  +  a  log  T10     .     .     .    (344) 


e 

90° 

80° 

70° 

60° 

50° 

40° 

30° 

20° 

10° 

0° 

19000 

-5.383 

-5.416 

-5.386 

-5.380 

-5.381 

-5.378 

-5.361 

-5.387 

-5.419 

-5.433 

18000 

-5.404 

-5.434 

-5.408 

-5.406 

-5.404 

-5.404 

-5.387 

-5.409 

-5.426 

-5.449 

17000 

-5.423 

-5.445 

-5.431 

-5.426 

-5.418 

-5.424 

-5.418 

-5.439 

-5.454 

-5.465 

16000 

-5.436 

-5.474 

-5.454 

-5.449 

-5.444 

-5.447 

-5.435 

-5.454 

-5.469 

-5.480 

15000 

-5.459 

-5.486 

-5.475 

-5.468 

-5.460 

-5.466 

-5.477 

-5.463 

-5.481 

-5.491 

14000 

-5.481 

-5.503 

-5.496 

-5.489 

-5.484 

-5.489 

-5.490 

-5.496 

-5.504 

-5.500 

13000 

-5.503 

-5.522 

-5.520 

-5.513 

-5.494 

-5.504 

-5.503 

-5.518 

-5.521 

-5.510 

12000 

-5.526 

-5.526 

-5.542 

-5.538 

-5.510 

-5.523 

-5.507 

-5.505 

-5.507 

-5.515 

11000 

-5.559 

-5.571 

-5.56'0 

-5.544 

-5.571 

-5.540 

-5.514 

-5.510 

-5.513 

-5.518 

10000 

-5.578 

-5.580 

-5.589 

-5.560 

-5.547 

-5.531 

-5.512 

-5.511 

-5.513 

-5.519 

9000 

-5.606 

-5.601 

-5.595 

-5.589 

-5.569 

-5.552 

-5.526 

-5.511 

-5.513 

-5.519 

8000 

-5.614 

-5.611 

-5.596 

-5.603 

-5.563 

-5.555 

-5.524 

-5.514 

-5.517 

-5.524 

7000 

-5.622 

-5.618 

-5.608 

-5.608 

-5.576 

-5.556 

-5.557 

-5.524 

-5.526 

-5.531 

6000 

-5.628 

-5.626 

-5.610 

-5.607 

-5.577 

-5.558 

-5.536 

-5.532 

-5.537 

-5.543 

5000 

-5.637 

-5.627 

-5.615 

-5.615 

-5.581 

-5.564 

-5.547 

-5.531 

-5.527 

-5.536 

4000 

-5.638 

-5.635 

-5.622 

-5.600 

-5.589 

-5.574 

-5.534 

-5.537 

-5.542 

-5.548 

3000 

-5.656 

-5.646 

-5.628 

-5.616 

-5.588 

-5.561 

-5.546 

-5.554 

-5.555 

-5.561 

2000 

-5.664 

-5.652 

-5.634 

-5.614 

-5.597 

-5.582 

-5.561 

-5.588 

-5.546 

-5.549 

1000 

000 

The  minus  sign  affects  only  the  characteristic  log  c  =  —  5.664. 

c  =  4.61  X  10  ~ 

TABLE  42 
THE  EXPONENT  a  IN  Kw  =  c  Ti0a    .     .     .     (344) 


z 

90° 

80° 

70° 

60° 

50° 

40° 

30° 

20° 

10° 

0° 

19000 

3.740 

3.755 

3.741 

3.739 

3.739 

3.738 

3.730 

3.742 

3.757 

3.763 

18000 

3.750 

3.763 

3.752 

3.750 

3.750 

3.749 

3.742 

3.752 

3.764 

3.770 

17000 

3.758 

3.768 

3.762 

3.760 

3.756 

3.759 

3.756 

3.766 

3.772 

3.777 

16000 

3.764 

3.781 

3.772 

3.770 

3.768 

3.769 

3.764 

3.772 

3.779 

3.783 

15000 

3.775 

3.786 

3.782 

3.778 

3.775 

3.778 

3.782 

3.776 

3.784 

3.789 

14000 

3.784 

3.794 

3.791 

3.788 

3.785 

3.788 

3.788 

3.791 

3.796 

3.793 

13000 

3.794 

3.803 

3.802 

3.798 

3.790 

3.794 

3.794 

3.801 

3.802 

3.797 

12000 

3.804 

3.804 

3.812 

3.810 

3.797 

3.803 

3.796 

3.795 

3.796 

3.799 

11000 

3.820 

3.825 

3.820 

3.813 

3.825 

3.811 

3.799 

3.797 

3.799 

3.801 

10000 

3.828 

3.829 

3.833 

3.820 

3.814 

3.807 

3.798 

3.798 

3.799 

3.801 

9000 

3.841 

3.838 

3.835 

3.833 

3.824 

3.816 

3.804 

3.798 

3.799 

3.802 

8000 

3.844 

3.843 

3.836 

3.833 

3.822 

3.818 

3.804 

3.799 

3.800 

3.804 

7000 

3.848 

3.846 

3.841 

3.833 

3.823 

3.818 

3.818 

3.804 

3.804 

3.807 

6000 

3.850 

3.850 

3.842 

3.831 

3.828 

3.819 

3.809 

3.807 

3.810 

3.812 

5000 

3.854 

3.850 

3.844 

3.837 

3.829 

3.822 

3.814 

3.807 

3.805 

3.809 

4000 

3.855 

3.854 

3.848 

3.838 

3.833 

3.826 

3.808 

3.810 

3.812 

3.814 

3000 

3.863 

3.858 

3.850 

3.845 

3.832 

3.820 

3.814 

3  .  817 

3.818 

3.820 

2000 

3.867 

3.862 

3.853 

3.844 

3.836 

3.830 

3.820 

3.832 

3.814 

3.815 

1000 

000 

The  exponent  a  -  3.50  in  dry  air;   a  =  4.00  in  a  perfect  radiator. 


THERMODYNAMIC   TABLES 


131 


V) 

d 
1 
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00  00  TH  t^.  TH 

IO   tO   TH   TH    O 

CM  CM  CO  CM  O 
CO  CO  CO  CO  CO 

II   II   1 

O  O  TH  o  t> 

TH   T-H   00  CO   CO 

777  i  i 

t>  CO  CO  CM  CO 

00  t^  <0  T)«  rH 

0 

O 

Oi  TH  CO  CM  to 
i—  1  CO  TH  TH  CO 
CO  CO  CO  CO  CM 

II   II   1 

77  ii  i 

TH  CO  t»  00  b- 

CT.  CO  00  TH  O> 

°0 

T—  1     tO     TH     CO     TH 

CO  TH  TH  CO  CO 
CO  CO  CO  CO  CM 

II  II  1 

gggSS; 

777  i  i 

°o        2 

CM 

£ 

00  TH  CM  O  CO 
O5  Ci  1>  to  CM 
CO  CO  CO  CM  CM 
II       II       1 

TJH  CO  CO  t*  lO 

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o           2 

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co           w 

i 

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II      II      1 

to  0>  0  CD  0 

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10  OJ  TH  «  0* 

7  i  ii  i 

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CO  t^  CO  CM  CM 

CM   T^    TH    IO   Tf 

CO  CO  CO  CO  CO 

II   II   1 

0  to  0>  CD  i-l 

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0             * 

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5 

O  CM  CO  CO  CO 
CO  CO  CO  CO  CO 

TH  CO  to  CO  CO 
CO  t>-  !>•  O  00 

7777  i 

1. 

10  CM  05  TH  TH 

t^.  o  TH  to  oo 

fe         w 

t,     » 

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co  co  co  co  co 

II   II   1 

OO  CO  0^  TH  CO 
tO  to  t»  TH  CM 

CO     CM     TH     TH     TH 

Mill 

CO  l>  CM  CM  00 

CO  CM  CO  TH  O5 

b 

00  !>•  CO  to  TH 
CM  CM  CO  CO  CO 

II   II   1 

77777 

CO  CM  «O  CM  tO 

00  CO  CO  CO  CO 

o 

8 

CM  CO  O5  CM  t^ 
00  O^  ^^  CM  CO 
CM  CM  CO  CO  CO 

Mill 

to  O5  CO  O5  CO 

TH    CO   O5   00   *>• 

CO     CM     TH     TH     TH 

II  II  1 

N 

O5  00  t>-  CO  to 

TH  CO  CM  TH  O 

132 


THERMODYNAMIC   METEOROLOGY 


0  0  CO  r^  rH 

CO  <M  iO  O  00 

rH   CO   CO   O5   <N 

T-H  CD  CO  00       • 

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GO  !>•  O5  O  O5 

rH  t>-  iO  CO  O5 

l>-   CD   CO   O5   CO 

rH    CO    (M    rH    O5 

O  C^  to  !>•  iO 

O  O  (M  l>-      • 

CO  CO  t^- 

T-H  00  to 

M    M    1 

M    M    1 

Mill 

i   M   i    : 

1   1   1 

rH   (M    O   00   CO 

GO   T-H    O5   CO   t* 

Oi  CO  TH  CN!  CO 

s  ^  §  g  g 

O  CO  CO  tO  (M 
CO  0i  1s**  1s*  CO 

GO  00  t^  ^«'     • 

co  co  i>  e*     ' 

00  O  00 

co  oo  ^ 

M    M    1 

M    M    1 

M    M    1 

MM: 

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<NTi«^<N  CO 

O  O  OS  t-  iH 

00  O  00  CO  00 

t-t-(NlO      • 

<N  CO  (N 

§82^SS? 

t2*o»co8 

t*  CO  r^  (M  t^ 
t»  T-H  rJH   CO  CO 

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SEVERAL  IMPORTANT  CONCLUSIONS  133 

The  Case  Ii  of  low  temperatures  in  the  isothermal  region 
shows  more  rapid  radiation  than  Case  L  of  high  temperatures, 
but  both  indicate  about  twice  as  much  radiation  as  in  the  con- 
vectional  region  C.  There  are  two  special  sources  of  heat,  (1) 
in  the  upper  cirrus  region,  and  (2)  at  the  surface  of  the  earth, 
or  in  the  lower  cumulus  region.  The  incoming  radiation  of  the 
sun  is  divided  into  two  nearly  equal  parts,  the  effective  radiation 
penetrating  to  the  surface,  but  diminishing  in  intensity  in  and 
below  the  cirrus  region,  30  to  10  kilometers,  and  the  returning 
radiation  neutralizing  an  equal  amount  of  the  solar  radiation. 
Pyrheliometer  observations  at  the  surface  give  2.00  calories  per 
square  centimeter  per  minute,  and  the  bolometer  observations 
probably  require  the  solar  constant  of  4.00  calories.  Hence 
2.00  calories  are  penetrating  to  the  surface  and  2.00  calories  are 
returning  to  space. 

Several  Important  Conclusions 
From  the  formulas, 


g     *   -   *0      =     -  ~  -    <?0        -  -  , 

g  Pio  (2i  -Z0)  =  -  (Pi  -  Po)  ~  J  Pio  (?i2  -  ?o2)  -  Pio  (Qi-Qo), 
we  must  lay  down  several  propositions. 

(1)  There  is  properly  no  such  a  thing  as  purely  dynamic 
meteorology,  which  is  defined  as  a  balance  between  the  three 
terms, 


g 


pio 


since  these   conditions  are  fulfilled  in  nature  only  under  tem- 
porary circumstances. 

(2)  In  Mar  gules'  paper,  "Ueber  die  Energie  der  Sturme," 
Jahrbuch   der   K.    K.    Central-Anstalt   fur   Meteorologie   und 
Erdmagnetismus,    Wien,    1903,    the    formulas    are    nearly    all 
adiabatic,  the    gas  coefficient  R  and  the  specific  heat  Cp  are 
constant,  so  that  the  radiation  term  (Qi  —  Qo)  cannot  be  com- 
puted, and  there  is  no  balance  possible  among  the  other  terms. 

(3)  In  V.   Bjerknes'   paper,    "  Dynamic    Meteorology   and 


134  THERMODYNAMIC  METEOROLOGY 

Hydrography,"  Carnegie  Institution  of  Washington,  D.  C.,  1910, 
the  density  is  computed  by  formula  (175)  instead  of  by  (176), 
and  it  is,  therefore,  a  mixed  system,  since  the  pressure  depends 
upon  (172),  and  R,  Cp,  are  taken  constant,  so  that  there  is  no 
theoretical  circulation  and  radiation  to  be  computed. 

(4)  In  Gold's  paper,  "The  Isothermal  Layer  of  the  Atmo- 
sphere and  Atmospheric  Radiation,"  Proceedings  of  the  Royal 
Society,  A,  Vol.  82,  the    assumption  is  made  that  the  mass  is 
proportional  to  the  pressure.     This  omits  the  important  terms 
I  PIO  (<?i2  —  <?o2),    PIO  (Qi  —  Qo),   involving    the   circulation   and 
radiation,  so  that  the  dependent  formulas  are  not  properly  sup- 
ported, because  the  adiabatic  case  is  in  reality  assumed. 

(5)  There  is  a  very  large  literature  in  meteorology  based  upon 
attempts  to  make  a  balance  of  the  equation, 

P  -  -  P 

g  (zi  -  *0)  =  -  -          -  -  i  (?i2  -  ?o2), 

Pio 

but  it  is  in  reality  without  substantial  importance.  In  spite 
of  the  serious  difficulty  that  exists  in  determining  the  (Qi—Qo) 
term,  it  is  necessary  that  this  should  be  done.  There  are  several 
large  observatories  for  balloon  and  kite  ascensions  which  record 
pressure  and  temperature,  but  not  humidity  and  wind  velocity, 
and  these  entirely  fail  of  their  purpose  in  advancing  the  interests 
of  meteorology. 

(6)  In  spite  of   the  fact  that  only  thermodynamic  meteor- 
ology can  have  any  permanent  value  in  science,  there  are  yet 
many  subordinate  problems  in  the  atmosphere  which  are  to  be 
studied  without  the  heat  term,  such  as  the  stream  lines  of  the 
circulation,  however  their  forces  may  have  been  developed.     Ac- 
cordingly, we  proceed,  under  dynamic   meteorology,  to  deduce 
the  general  equations  of  motion,  in  the  rectangular,  the  cylin- 
drical, and  the  polar  co-ordinate  systems,  together  with  several 
minor  terms,  in  order  to  study  their  application  in  local  storms 
and  circulations  of  various  types. 


CHAPTER  III 

The  Hydrodynamics  of  the  Atmosphere 
The  Co-ordinate  Axes 

THE  general   equations   of  motion  will  be  assumed  from 
hydrodynamics,  because  they  are  well  known,  and  the  proof  is 


Zenith 


y  / 


South 


x,  y,  z  (Rectangular) 

accessible  in  many  treatises,  but 
the  equations  needed  in  meteor- 
ology will  be  deduced  from  them 
as  briefly  as  possible.  There  are 
three  systems  of  co-ordinates,  the 
rectangular,  the  cylindrical,  and 
the  polar,  as  represented  in  the 
diagrams  of  Fig.  14. 

Starting  at  the  point  0  as  the 
origin  of  co-ordinates,  one  can 
reach  the  point  P  directly  along 
the  line  0  P,  or  indirectly,  in 
rectangular  co-ordinates  along  the 

distances  x,  y,  z,  in  succession  parallel  to  the  axes;  in  cylindrical 
co-ordinates  along  the  radius  or,  through   an   angle  <f>  counted 

135 


r,  \  B  (Polar) 


FIG.  14.     Three  systems  of  co- 
ordinate axes.    Compare  Fig.  2. 


136  THE  HYDRODYNAMICS   OF   THE  ATMOSPHERE 

from  an  initial  line  of  reference  where  0  =  0,  and  along  z  to 
P;  in  polar  co-ordinates  along  a  line  r  whose  position  in  space 
is  determined  by  the  angle  A  in  the  plane  x  y  counted  from  an 
initial  line  at  the  axis  of  x  on  that  plane,  and  in  the  plane  z  v 
at  the  angular  distance  6  from  the  axis  of  rotation  z.  These 
systems  are  convenient  in  different  problems  and  must  each 
be  developed. 

The  Co-ordinate  Velocities 

If  q  is  the  velocity  along  the  line  O  P  with  which  a   mass 
is  moving,  the  co-ordinate  velocities  are  as  follows: 

Rectangular  Cylindrical  Polar 

(352)  u  =  -jfi      (353)  u=^'          (354)  u  -  r^y 

dy  d<t>  d^ 

v  =  dT          "*'*Tf  V  =  rsm9d-i 

• 

dz  dz  dr 


The  Co-ordinate  Accelerations 


d  t       d  t2  d  ' 


dw       d*z  .        dzz  .     '  dzr 


The  Constituents  of  the  Force  in  Any  Direction 

Force  is  measured  by  the  acceleration  of  a  mass  in  any 
direction,  because  it  takes  force  to  change  the  velocity  of  the 


THE   CONSTITUENTS   OF   THE   FORCE   IN  ANY  DIRECTION    137 

mass  m  at  a  given  point.  This  will  be  taken  as  unity,  m  =  1,  in 
the  preliminary  equations.  A  mass  of  gas  or  liquid  can  undergo 
changes  in  inertia,  changes  in  volume  by  expansion  or  contrac- 
tion, and  changes  in  figure  by  rotation,  and  these  three  types  of 
forces  must  be  placed  in  the  general  equations  of  motion.  The 
causes  that  produce  these  forces  in  the  atmosphere  of  the  earth 
are  external  and  internal.  The  external  forces  are  due  to 
gravitation  or  the  potential  changes  with  position;  the  internal 
forces  are  due  to  pressures  which  vary  in  different  directions, 
and  cause  motion  during  the  restoration  to  normal  equilibrium. 
The  primary  source  of  these  pressure  forces  is  the  distribution  of 
the  thermal  energy  derived  from  the  solar  radiation,  or  trans- 
ported in  currents  of  circulation. 

The  Force  of  Inertia 

This  is  a  partial  differential  of  the  velocity  in  the  direction 
of  the  co-ordinate  axes,  and  is 

du     dv     dw 


in  each  of  the  co-ordinate  systems. 

The  Forces  of  Expansion  or  Contraction 

It  is  convenient  to  express  the  differentials  of  the  linear  dis- 
placements in  each  co-ordinate  system,  referred  to  the  x,  y,  z 
system. 

Rectangular                  Cylindrical  Polar 

(359)     3  #  =  3  #•  (360)     3  x  =  3  w.    (361)  3  x  =  r  3  fr 

dy  =  d  y>                dy  =  ttr  30.  3  ?  =  r  sin  03  A. 

82  =  32.                 32  =  32.  3^  =  3  r. 

Using  these  values  of  3  x,  3  y,  3  2,  we  obtain  for  the  accelera- 
tions due  to  expansion  or  contraction: 


138 


THE   HYDRODYNAMICS    OF    THE   ATMOSPHERE 


Rectangular 

,      x      3#         du          3w 

(362)  u—  +  v^-  +  w—. 

dx         dy          82 

3fl         dv          dv 

u  —  +  v  -r — 1-  w  ^— . 
80;         dy         32 


u 


Cylindrical 

dv 


3  z> 
v  —  - 


dw 


^— 
dy 


. 

dz 


—  —  +  V 


to-  90 


dz' 

d~z 
dw 
d~z 


(364)^+,-, 


du 


32; 


dw  dw  dw 

U h  V ; ~  +  W . 

rd&        r  sin  0  3  A          3  r 

In  order  to  give  some  practical  idea  of  the  meaning  of  these 
terms,  the  following  example  is  taken  from  the  Cottage  City 
water-spout,  without  explanation. 

TABLE  44 
THE  RADII  IX  AND  VELOCITIES  «,  v,  w,  IN  (1)  OUTER  AND  (2)  INNER  TUBES 


Height 

Radius  W 
(1)          (2) 

Velocity  u 

(1)              (2) 

Velocity  v 
(1)          (2) 

Velocity  w 
(1)          (2) 

az  90° 

83.3 

51.9 

0 

0 

11.52 

18.49 

0.77 

1.98 

80° 

84.0 

52.3 

-  2.00 

-  3. 

21 

11.34 

18.20 

0.76 

1.95 

70° 

86.0 

53.6 

-  3.94 

-  6. 

32 

10.82 

17.37 

0.72 

1.86 

60° 

89.6 

55.8 

-  5.76 

-  9. 

24 

9.97 

16.01 

0.66 

1.71 

50° 

95.2 

59.3 

-  7.40 

-11. 

88 

8.82 

14.16 

0.58 

1.52 

40° 

103.9 

64.8 

-  8.82 

-14. 

16 

7.40 

11.88 

0.50 

1.27 

30° 

117.9 

73.4 

-  9.97 

-16. 

01 

5.76 

9.24 

0.38 

0.99 

20° 

142.5 

88.8 

-10.82 

-17. 

37 

3.94 

6.32 

0.26 

0.68 

10° 

200.0 

124.6 

-11.34 

-18. 

20 

2.00 

3.21 

0.13 

0.34 

0° 

00 

00 

-11.52 

-18. 

49 

0 

0 

0 

0 

It  is  seen  that  the  velocities  in  meters  per  second  in  the 
lower  half  of  the  dumb-bell  vortex  of  this  water-spout  undergo 
changes  from  the  vortex  tube  (1)  to  the  vortex  tube  (2),  and  that 
as  the  radius  w  and  the  height  az  change,  these  velocities  change 
by  certain  laws.  By  taking  the  differences  3w,  3»,  3w,  3  -or, 


THE  FORCES  OF  ROTATION  139 

or  3  0,  3  z  in  cylindrical  co-ordinates,  and  using  the  mean  values 
of  um,  vm,  wm,  the  forces  that  caused  these  changes  from  point 
to  point  in  the  vortex  can  be  computed. 

The  Forces  of  Rotation 

If  a  mass  moves  relatively  to  fixed  axes  without  rotation  there 
are  no  forces  other  than  inertia  and  compression  or  expansion. 
If  the  mass  also  rotates,  a  new  set  of  forces  of  rotation  is  intro- 
duced which  may  be  analyzed  as  follows:  Assume  that  there  is 
a  set  of  rectangular  axes  fixed  in  space,  and  that  another  set  of 


4-zo>2 

VU1 


+y<»i 


FIG.  15.    Angular  velocities  of  motion  about  fixed  axes. 

axes  attached  to  the  rotating  body  moves  relatively  to  the  fixed 
axes  with  the  co-ordinate  angular  velocities  wi  about  the  axis 
x,  o)2  about  the  axis  y,  and  ^3  about  the  axis  z.  The  entire 
system  of  rotational  velocities  at  the  distances  x,  y,  z,  from  the 
origin  of  rotation  is  shown  on  Fig.  15. 

The  right-handed  rotation  is  defined  as  that  in  which  a 
right-handed  screw  is  turned  to  advance  while  the  axis  x  moves 
toward  y,  the  axis  y  toward  z,  the  axis  z  toward  x,  in  transla- 
tion along  z,  x,  y,  respectively,  and  all  co-ordinates,  of  fixed  and 
moving  axes,  are  so  related.  Thus,  in  rotation  about  z,  with 
the  angular  velocity,  ^3,  at  the  distance  x  there  is  instantaneous 


140       THE  HYDRODYNAMICS  OF  THE  ATMOSPHERE 

velocity  +  x  co3  parallel  to  the  axis  y}  and  at  the  distance  y 
there  is  instantaneous  velocity  —  y  co3  parallel  to  the  axis  x, 
that  is,  in  the  negative  direction.  The  same  considerations  give 
the  component  linear  velocities  parallel  to  the  axes. 

(365)  Parallel  to  the  axis  x,  —  y  co3  +  z  o>2, 
Parallel  to  the  axis  y,  —  z  coi  +  x  co3, 
Parallel  to  the  axis  z,  —  x  0)2  +  y  +  y  "i- 

These  symbols  are  all  arranged  in  the  cyclic  order,  and  are  easily 
verified  from  Fig.  15. 

Similarly  the  component  accelerations  are  found  by  substi- 
tuting u,  v,  Wj  for  x,  y,  z,  in  succession,  and  we  have  the  accelera- 
tions parallel  to  the  axes: 

(366)  Parallel  to  the  axis  x,  —  v  co3    +  w  co2, 
Parallel  to  the  axis  y,  —  w  wi  +  u  co3, 
Parallel  to  the  axis  z,   —  u  o>2  +  v  coi. 

Since  these  forms  are  entirely  general,  it  is  only  necessary 
to  substitute  the  special  values  of  «i,  co2,  co3,  for  given  cases,  to 
apply  the  formulas  to  particular  problems.  If  only  fixed  axes 
are  employed,  we  have  «i  =  0,  co2  =  0,  co3  =  0.  If  cylindrical 
axes  are  employed,  there  is  rotation  about  the  axis  z  only,  so 

u 
that   coi  =  0,    co2  =  0,  co3  =  -f-  —  .     If  polar   co-ordinates   are 


employed  the  angular  velocities  become  «i  =  --  ,  co2  =  + 

o>3  =  +  -     —  -.     Placing    these    results    in    tabular    form, 
Y  tan  u 

have,  as  can  be  seen  by  the  definition  of  angular  velocity, 


Rectangular 
Fixed  Axes 

Cylindrical 
Co-ordinates 

Polar 
Co-ordinates 

(367) 

coi  ==  0» 

(368)  «i  =  0. 

(369)  coi  =  -  -  |. 

C02    =    0. 
C03    =    0. 

co2  =  0. 
V 

w 

<*  =  +  7- 

y              l 

0)8       +  w' 

THE   FORCES    OF   ROTATION 


141 


In  cylindrical    co-ordinates    the    angle    increases   with    the 
angular  velocity  ^3  =  ^*     In  polar  co-ordinates   the   angular 

velocities  may  be  illustrated  by  Fig.  16. 

Let  a  mass  move  from  P0  to  P3  relative  to  the  rotating  earth, 
whose  axis  of  rotation  is  0  Z.    Lay  down  the  fixed  axes  x,  y,  z, 


Pole 


FIG.  1 6.    Angular  velocities  of  moving  axes  relative  to  fixed 


at  the  center  O,  draw  the  radius  r  to  PQ  at  the  polar  distance  0, 
in  longitude  A  counted  from  an  initial  meridian.  The  angular 
velocity  of  the  rotating  earth  is  ^3;  r  sin  6  is  the  perpendicular 
distance  of  PQ  from  0  Z,  and  r  tan  6  is  the  tangential  distance  from 
PQ  to  O  Z.  If  the  mass  moves  from  PQ  to  PI  it  rotates  about  the 

u 
axis  y  with  the  angular  velocity  co2  =  -,  since  it  moves  in  the 

positive  direction  of  x;  if  it  moves  from  P0  to  P%  it  rotates  about 

v 
z  with    the  angular  velocity  o/3  =  rzrr^J    ^  ^  moves  from 


rtan0 


142 


THE  HYDRODYNAMICS  OF  THE  ATMOSPHERE 


to 


P2  it  rotates  about  x  with  angular  velocity  —  coi  =  -, 


because  in  the  motion  from  P0  to  PZ  the  velocity  is  in  the  negative 
direction  of  rotation.  If  the  mass  moves  from  P0  to  P3  along  the 
radius  there  is  no  rotation.  These  rotational  relations  can  be 
rigorously  proved  by  analytical  demonstration,  but  as  the  analysis 
is  rather  complicated  the  reader  is  referred  to  the  standard 
treatises  by  Basset,  Lamb,  W.  Wien,  and  others.  The  formulas 
for  the  angular  velocities  in  terms  of  the  linear  velocities  are  as 
follows : 

(370)    Rectangular  co-ordinates.     2  coi  =  TT—  —  — . 


(372)     Polar  co-ordinates. 


8  u       'dw 

o ^  

8  z       8#* 
8  v       du 

2  COs    —  7T         —   ~^      . 

8*       dy 


(371)     Cylindrical  co-ordinates.       2 


co2 


8  co   -     8  v 
iff  8</>       8  z 

du       8_co 
8-or* 


8  v        du         v 
2  cos  =  ~ ^  +  — • 


2coi  = 


8  co  8  v 

r  sin  6  8  A       8  r 


du        dw 
2  co2  =  ^-  -  -^-2 


2  0)3   = 


8  v  8  u 

r'dB  ~~  r  sin  0  8 


r  tan 


If  the  linear  velocities  u,  v,  co,  at  the  point  x,  y,  z,  are  known, 
the  angular  velocities  coi,  co2,  co3,  can  be  computed  in  the  three 
systems  of  co-ordinates. 


THE  PRESSURE  GRADIENTS  143 


It  should  be  observed  that  since, 

v  .  v  z;cos0 

"3  =  rlfo*'  and  °"  =  TtoTo  =  ,-» 

(373)  0/3  =  o)3  cos  0, 

so  that  at  the  pole  for  0  =  0,  cos  0  =  1,  o/3  =  co3,  and  at  the 
equator  for  0  =  90°,  cos  0  =  0,  o/3  =  0.  At  the  pole  a  point 
rotates  about  the  axis  of  rotation  with  the  angular  velocity  of 
the  rotating  earth,  but  at  the  equator  it  does  not  have  angular 
velocity  about  z. 

The  Pressure  Gradients 

The  earth  in  its  rotation,  acted  upon  by  the  gravitation  of 
its  own  mass,  has  assumed  the  form  of  an  oblate  spheroid,  being 
flattened  at  the  poles,  and  if  the  atmosphere  were  not  acted 
upon  by  thermal  forces  it  would  be  arranged  in  layers  of  density 
parallel  to  those  of  the  earth's  solid  mass.  The  heating  of  the 
atmosphere  in  the  tropics  by  solar  radiation,  and  secondarily  in 
the  other  latitudes,  disturbs  these  level  surfaces  of  pressure,  by 
lifting  some  areas  and  depressing  other  areas  during  the  processes 
of  heating,  cooling,  and  circulation.  The  change  in  pressure 
from  the  normal  pressure  in  a  given  linear  distance  is  the  pressure 

gradient  —  -7-,  and  since  the  forces  are  directed  from  the  higher 

to  the  lower  pressure  the  minus  sign  is  used,  —  -7-.     Finally,  the 

forces  are  all  to  be  reduced  to  the  unit  density  so  that  the  forms 
become,  for  the  three  co-ordinate  systems,  respectively, 

Rectangular  Cylindrical  Polar 

Co-ordinates  Co-ordinates  Co-ordinates 


(374)     -  (375)     -  (376)     - 

- 


. 

3  x  p  a-or  pr  30 

ap  iap  iap 


py  p  t*r< 

iap        iap        i ap 

p  dz'  p  dz'  p  a  r * 


144  THE   HYDRODYNAMICS   OF   THE   ATMOSPHERE 

These  expressions  can  be  evaluated  into  many  forms,  which  are 
convenient  for  practical  computations,  as  will  be  shown  in  a  later 
section. 

The  Potential  Gradient 

If  the  potential  of  the  external  forces  of  attraction  of  the 
earth's  mass  is  V,  then  the  forces  due  to  such  a  potential  are, 


8  V  8  V 

In  the  case  of  the  earth's  mass  the  forces  —  -r—  and  —  -^—  , 

dx  dy 

on  the  meridians  and  on  the  parallels  of  latitude,  respectively, 
are  chiefly  concerned  with  the  determination  of  its  existing 
figure.  In  the  problems  of  meteorology  these  forces  can  be 
neglected,  so  that  there  remains  only  the  vertical  potential 
gradient, 

(378)  -!£--* 

There  has  been  considerable  confusion  in  the  literature  of 
this  subject  because  the  positive  direction  is  taken  upward 
by  some  authors,  but  downward  by  others.  If  the  positive 
direction  is  upward,  as  where  the  positive  motion  along  the 
radius  is  outward  from  the  center  of  the  earth,  we  have 

(379)  —^  =  +  g,  and  V  =  +  g  z. 

If  the  positive  direction  is  inward,  as  along  the  path  of  a 
falling  body,  we  have, 

d  V 

(380)  -j^  =  -  g,  and  V  =  -  g  z. 

The  positive  direction  upward  is  used  by  Ferrel,  Sprung, 
Oberbeck,  Basset  in  some  sections,  Helmholtz,  Bigelow;  the 
positive  direction  inward  is  used  by  Basset  in  some  sections, 
Lamb,  V.  Bjerknes. 

The  force  of  gravity  is.  however,  modified  on  the  rotating 


EQUATIONS   OF   CONTINUITY  145 

earth  by  the  centrifugal  force,  which  acts  only  in  planes  per- 
pendicular to  the  axis  of  rotation.  These  forces  can  be  resolved 
along  and  perpendicular  to  the  axis,  as  follows,  since  the  centrif- 
ugal force  is  J  co02  w2. 

The  total  potential  is, 

(381)  V  =  g  r  +  i  coo2  w2, 

where  co0  is  the  angular  velocity,  and  co0  =  n  in  the  notation 
for  the  rotating  earth.  By  (66)  we  have 

R2 

(382)  g  =  go— ,  and 

R2 

(383)  'gr  =  g0—,  so  that 

(384)  V 

Taking  the  differential  along  the  axis  z,  and  perpendicular 
to  it  along  -or,  there  results: 

dV  R2  dr 

-77 -+«?-*;' 

dV  R2  dr 

(386)  -^  =  +so-^-"2o*r. 

We  have: 

(387)  r2  =  x2  +  y2  +  z2  =  -or2  +  z\    Differentiating, 

(388)  2rdr  =  2wd'&+2zdz,  and 

(389)  Tz=7*      £=7'   Hence>for*  =  >* 

(390)  -  -^  =  g  '  j  =  g  cos  e. 

(391)  -  -7—  =  g  '  —  -  co?2  w  =  g  sin  0  -  co32  or. 

a  w  Y 


Equations  of  Continuity 

When  a  mass  of  air  streams  through  a  given  space,  as  a  cubic 
meter  at  a  given  place,  as  much  air  must  pass  out  of  it  as  enters 


146  THE  HYDRODYNAMICS   OF   THE  ATMOSPHERE 

it,  or  else  there  will  be  congestion,  and  a  change  in  the  con- 
tinuity. The  equations  which  finally  control  any  solution  of 
current  functions  must  satisfy  these  equations  of  continuity. 

(392)    Rectangular     du       8_y       dw  = 
Co-ordinates,     ox  ~ 

(393) 

(394)  Cylindrical 
Co-ordinates. 

(395)  Polar  f  9  («  sin  6)  +  f  dv  +  ^  $ 3  (>•«>)  =  Q 
Co-ordinates.  o  0  8  A  8  r 


Operator  V2 

The  sum  of  the  second  differentials  in  three  co-ordinates  is 
often  used  in  analytical  discussions,  and  the  symbol  V2  has  been 
adopted  for  this  process. 

(396)  Rectangular    y2  =      82   _,     92    _,     82 
Co-ordinates.          '   3*.      8/      ,8*" 

(397)  Cylindrical    y2=   _?L_,J__3_  +  1     92     ,     S2 
Co-ordinates.  3^2       tcr  8-ar       -or2  8  ^2        8  22' 


(398)  Polar  y.'^J!. +ljL.  +  lI  _*.-.,. 2*£ 4- + 

Co-ordinates.  dr2       r  dr       r2  8  0*         r2    d  0 

1       _^_ 
r2  sin2  0  8  A2' 

d 

jTAe  Toto/  Differential -r. 
at 

The  symbol -7^  is  often  used  to  include  the  terms  of  the 
inertia  and  expansion  or  contraction. 

(399)  A    =     3-  +  u   JL  +  v    Ar    w  A 

<//        8£          8ic          83;         82* 

There  are  a  series  of  complicated  terms  used  to  express  the 
internal  forces  caused  by  the  stretches,  the  shears,  the  dilatation, 
the  tractions  due  to  elasticity  and  viscosity  within  the  masses, 
but  these  will  be  omitted  in  this  place.  They  are  summarized 
on  pages  499-501  of  the  Cloud  Report. 


SUMMARY    OF   EQUATIONS    OF    MOTION  147 

Summary  of  the  Equations  of  Motion 

By  putting  together  the  terms  that  have  now  been  explained, 
there  result  the  general  equations  of  motion  which  are  at  the 
basis  of  all  the  dynamic  meteorology,  omitting  the  heat  term  d  Q. 

Rectangular  Co-ordinates 

/,™\          18P       du          du        du   t       du 
(400)    --  •=—  =-^T:+W—  +v—  +  w  —  —  a  co3  +  w  w2. 
p  dx        dt          dx        d  y          dz 

1  8P       dv  dv         dv  dv 

-Pdj=^i+u-d'x+vd-y  +  wd-z-w"l+ua3- 
1  8P       dw         dw         dw          dw 


Cylindrical  Co-ordinates 

,      .         18P8w  8w  8w  du 

(401)    --  -—  =  —  +  «  ^—  +  v  —  r-  +  w  ^— 
P  8zD"       3/  dv-        ttdp  82 

^  °° 
"JJ         1    8P       8^ 


-,  +  «  o  --  r-  »  —  ^~  +  w  ^— 
^  »         P  -nr8^      8^  9w         WO^P  80 

^"3          1  8P       dw  dw  dw  dw 

^  .5       —  --   =  -  -U  11    -  -\-  71  -   -I-  7£i   -- 

^ 


Co-ordinates 
8P          8w          8w 


•s  r  r 

g«          1       8P  dv  dv  dv  dv 


uv  t  wv 

—  cot  0  +  —  . 


_      ^          _  __  _ 

"  ''^    U~*  V  dl^    W3r  ~ 

u2       v2 


148  THE  HYDRODYNAMICS   OF   THE  ATMOSPHERE 


Equations  of  Motion  for  the  Rotating  Earth 

The  equations  of  motion  for  moving  axes  are  further  modified 
when  the  axes  are  attached  to  the  earth  which  is  rotating  with  the 
constant  angular  velocity  ojs. 

Cylindrical  Co-ordinates  on  the  Rotating  Earth 

The  linear  velocities  remain  the  same  except  that  the  linear 
velocity  eastward  is  increased  by  the  term  w3.ttr  cos  6.  The 
angular  velocity  about  z  is  changed  by  the  addition  of  the  term 

co3   COS    6. 

(403)     Linear  velocities,   u'  =  u. 

i)r  =  v+(*>3 .  IB  cos  e. 

w'  =  w. 
Angular   velocities,  a/i  =  0. 

«/2    =    0. 


6  H . 

iff 


The  partial  differentials  are  also  modified: 


8  z/       8  (v  +  o)3.  ttr  cos  0)        dv   , 

•97--        ^T  •;-9F  +  «.«.cos». 

8  w' 


Substituting  these  terms  in  the    general  equations   (400), 
and  using  (399),  we  have: 

(405)     -  —  ^—    =  -r!  -  (ojs.-ar  cos0  +  fl)(co3cos  0  +  — ). 
p  ow         at  \  TX J 

1    dP       dv  /  v\ 

^T  =  -r*  +  u- ^  cos  0  +  u      co3  cos  0  +  —  ) . 

p  wo$      at  \  TX J 

_1_8^P         dw 
'  p  dz      =  dt  +g' 


POLAR  CO-ORDINATES   ON  ROTATING  EARTH  149 

Performing   the   algebraic  work,   and   substituting  for   the 

v 
relative  angular  velocity  eastward  v  = — ,  we  find: 

1  9P         du  v2       du 

(406)     —  —  -ZT-     =  -77  —  2  o>3  cos  B.v  —  —   =  -77  — 
p  Qfff         a  t  tff        a  t 

f^  (2o?3  cos  B  -f-  v)  v. 

i 
V-§  1    9P        dv  uv       dv 

z* 5^  =  Ti  +  2a>3  cos  O.u  +  —  =  -T-,  + 

p  tcrc/0       d  f  *&       a  i 

(2  o>3  cos  0  -f  v)  w. 

_1_^P_^  dw 

'p92       ~  dt       S  '   dt       S' 


Co-ordinates  on  the  Rotating  Earth 

The  linear  velocity  eastward  is  increased  by  the  term  o>3  r  sin  6, 
and  consequently  the  angular  velocities  o>i  and  co3  are  modified 
to  conform  with  it. 

(407)  Linear  velocities,  southward,  uf  =  u. 

eastward,  v'  =  v  +  co3  .  r  sin  6. 
zenithward,  w'  =  w. 

v  +  co3  r  sin  0 

(408)  Angular  velocities,  about  the  axis  re,  a?  i  =  —  -     -   —  . 


axis  y,  0/2  =  +  —  . 

,   v  +  co3  r  sin  B 
axis  0,  w  i  = 


r  tan 


(409j    The  partial  differentials  ^—  =  ^--. 

u  t        o  t 


*dv'       9  (v  +  o>3  r  sin  0)  _  9  0 
F7  =  9/  =  97  " 

u  .  o)3  cos  0  +  w  .  w3  sin  B. 


d~t   =  97* 


150  THE   HYDRODYNAMICS    OF   THE   ATMOSPHERE 

Substituting  these  values  in  the  equations  (400),  we  have, 
1  8P  du 


1        8  P          d  v  /v  +  co3  r  sin 


rtane 


w — . 


or        _av  /v  +  cc3  r  sin  0\ 

7rsin08A  =  ~57~     *V r          J" 

/v  +  co3  r  sin  0\ 
M(      rung      )• 


/  -x 

— —          =  -77  —  w  .  —  —  (v  +  co3  r  sm  0) 

p  8  r  dt  r 

i)  -f-  o>3  r  sin  0 


/^  -T  co3 : 
\  r 


Performing  the  multiplications  and  reductions, 
18P  d  u      v2  uw 

dt        r  r 

2  co3  cos  0  »  .    .    .    —  r  co32  sin  0  cos  0. 

*S*  a  1          8P  dv         UV  WV 

2  co3  cos  0  .  w  +  2  co3  sin  0  .  w.   .    .   . 

^SP  _  dw  u*  +  v2  _ 

p  d  r  dt  r 

2  co3  sin  0  .  v   .    .    .    —  r  co32  sin2  0. 

The  terms  in  2  co3  represent  the  deflecting  forces  due  to  the 
earth's  rotation,  which  always  act  at  right  angles  to  the  linear 
velocity  q  with  which  a  mass  is  moving  in  any  direction.  They 
deflect  a  moving  body  to  the  right  in  the  northern  hemisphere, 
but  to  the  left  in  the  southern  hemisphere.  The  deflecting  force 
is  a  maximum  and  equal  to  2  co3  q  on  the  horizontal  plane  at  the 
poles;  it  is  equal  to  zero  at  the  equator,  for  all  velocities  in  the 
horizontal  plane.  If  the  velocity  is  vertical  the  deflecting  force 
due  to  this  term  is  zero  at  the  pole  and  equal  to  2  cos  w  at  the 
equator.  The  terms  in  r  co32  represent  the  forces  which  change 
the  figure  of  the  earth  from  a  sphere  to  an  oblate  spheroid. 


CONNECTION  BETWEEN  THE  GENERAL  EQUATIONS       151 

They  need  not  be  considered  in  practical  meteorology,  but  are 
important  in  geodesy. 

The  derivation  of  the  general  equations  of  motion  given 
above  is  exceedingly  simple  and  direct,  showing  immediately 
where  all  the  terms  come  from  which  are  concerned  with  gen- 
eral motions.  They  are  equally  true  for  the  sun,  the  earth,  and 
the  planets,  and  all  observed  motions  must  conform  to  them. 

Connection  Between  the  General  Equations  of  Motion  and  the 
Thermal  Equations  of  Energy 

We  found  that  with  changes  of  the  temperature  gradient 
the  variation  of  the  pressure  —  is  expressed  by  equation  (190), 

when  there  is  no  change  in  heat  Q  not  otherwise  accounted  for 
in  this  process  of  motion.  But  in  case  all  the  heat  energy  is 
not  expended  in  motion,  as  where  a  part  escapes  in  radiation  or 
in  internal  molecular  or  atomic  motions,  as  in  ionization,  a  new 
term  must  be  added  to  take  this  into  the  account,  so  that 
o  -p 

(412)  —  =  -J  +  nCpdTt+CpTlogT.dn. 

In  order  to  avoid  confusion  of  symbols  in  this  set  of  equations, 
take  v  =  o>3  the  angular  velocity  of  the  rotating  earth,  and 
combine  the  equations  (411)  and  (412).  It  should  be  noted 

v 

that  since  V  =  — : — -,  by  Fig.  16,  we  have 
r  sin  0' 

,      ^     v  v2  cot  0  uv  cot  6 

(413)  —  =  v  sin  0, =  v  cos  0  .  v,       =  u  cos  0  .  v, 

and  we  shall  make  use  of  them  in  the  following  transformations. 
In  order  to  connect  together  the  hydrodynamic  and  the  ther- 
modynamic  systems  by  the  law  of  the  conservation  of  energy,  we 

have  —  the  same  in  both  systems.       Combining  these  terms 
P 

and  making  the  reductions,  substituting  d  xt  dy,  d  z  for  the 
corresponding  expressions  of  the  linear  displacements  in  polar 
co-ordinates,  and  adding  d  Q  for  the  change  in  the  heat  contents, 


152  THE   HYDRODYNAMICS    OF    THE   ATMOSPHERE 

and  d  J  for  the  energy  in  the  form  of  electric  and  magnetic  forces, 
there  result  for  the  forces  of  acceleration, 

The  General  Hydrodynamic  and   Thermodynamic  Equations  of 

Motion 

,A1A.  1  dP  du  x         uw 

(414) — -  = cos  6  (2  co3  +  v)  v  + 

o  o  x  a  t  r 

^ 


-  g  —  =  -TT  +  cos  6  (2  w3  +  v)  u  +  sin  9(2  w3  + 


1  QP      dw 


Corresponding  with  these  co-ordinate  equations  is  the  differ- 
ential equation, 

dP  dq       dQ       dJ  ,    dT 

(415)      ~-  =  ^^        +         +-=-^-- 


This  equation  has  been  already  discussed  in  its  vertical 
variations,  but  the  more  difficult  task  is  to  explain  its  meaning 

in  the  horizontal  directions   (#,  y).      The   terms  -77,  -T-,  -7— 

U    /  U    /  U     / 

contain  the  inertia  and  the  forces  of  expansion  and  contraction 
as  expressed  in  equations  (358)-(364),  (396)-(398),  and  these 
must  generally  be  employed  in  the  study  of  tornadoes,  water- 
spouts, hurricanes,  ocean  and  land  cyclones.  The  terms  in  / 
can  hardly  be  discussed  until  the  subjects  of  absorption  in  the 
spectrum,  scattering  in  the  atmosphere,  electric  and  magnetic 


THE    EQUATIONS    FOR    THE    WORK    OF    CIRCULATION         153 

forces  can  be  more  thoroughly  worked  out.     The  difficulties  in 
determining  the  value  of  Q  have  already  been  stated. 


The  Equations  for  the  Work  of  Circulation 

If  the  equations  for  the  force  of  acceleration  (414)  are  multi- 
plied by  d  x,  d  y,  d  z,  respectively,  that  is,  if  the  force  is  multi- 
plied by  the  distance  through  which  it  acts,  they  give  the  work 
expended  in  transporting  the  mass  from  one  point  to  another. 
Still  retaining  the  unit  mass,  m  =  1,  the  work-equations  become, 

(415)     --  =  du-j-  —  cos  d  (2  co3  -f-  v)  v  d  x  -f 


---  =  dv  -77  +  cos  0  (2  co3  +  v)  udy  + 
p  at 


!      Q          d  y. 


3P  dz 

—  =  d  w  -j-.  —  sin  6  (2  oj3  -f  v)  v  d  z  — 
p  at 

uudz 


r  dz 

We  have  the  auxiliary  equations, 
(416)       v  d  x  =  u  d  y,     w  d  x  =  u  dz,     w  d  y  =  v  d  z, 

and  it  is  seen  that  by  substitution  all  the  terms  cancel  except 
the  following,  after  using  the  total  differential, 

—  - 

The  integral  equation  is  known  as  Bernoulli's, 

/cL  P  f* 

—  =  1  (W2  +  Z,2_|_W2)_|_  /  gdz  +  (Q  +  J),  one  limit, 

(419) — — -  =  |  (qi2  -  £o2)  +  g  (zi  -  z0)  +  (Qi  -  Qo)  + 

PIO 

(/i  —  /0),  two  limits. 


154       THE  HYDRODYNAMICS  OF  THE  ATMOSPHERE 

The  other  combination,  in  (414),  gives, 

(420)     -  f —  =  -  fnCpdT  +  f  Cp  T log  T d n.  as  (202). 
J     p  J  J 

In  case  /  —  J0  =  0,  we  have,  in  practice,  for  the  static 
pressure  P'  and  the  dynamic  pressure  P",  if  P  =  P'  +  P". 


(421)  —  /  =  g  (ZI—ZQ)  =  —  Cpni(Ti  —  T<?),  static  pressure, 

J      p 

(422)  -  f =  J  (gi2  -  g02)  +  (Qi  -  Qo),  circulation  and 

J      p 

radiation, 

//***  .A.  /"""*  A        \      /T*  T1    \  //"*  A.  /"^  t        \ 

=  w^a  ~~  Lpio)  (J-  a  —  J-  o)  —  n\  \Lpa  ~  ^pio) 

(T,  -  r0). 

The  equation  (421)  has  already  been  discussed  under  Barom- 
etry  (39)  with  others;  also,  under  gradients  (12)  with  others. 
The  equation  (422)  remains  to  be  considered  under  circulation 
and  the  variation  of  the  gradients. 


/dP 
. 
p 


There  are  several  methods  of  treating  the  computations 
involved  in  the  thermodynamic  and  static  equation  (419), 
which  considers  the  circulation  and  the  vertical  pressure  along 
with  the  thermodynamic  sources  of  the  pressure  differences. 
Since  all  the  terms  in  (415),  which  represent  the  deflecting  and 
the  centrifugal  forces,  disappeared  on  the  substitution  of  the 
values  in  (416),  it  follows  that  these  forces  are  always  acting  at 
right  angles  to  the  direction  of  the  motion  of  the  mass  moving 
with  the  velocity  q  in  any  direction  relatively  to  the  surface  of 
the  rotating  earth.  The  deflecting  and  the  centrifugal  forces 
have,  therefore,  no  effect  upon  the  work  of  the  circulation  of  the 
atmosphere,  any  more  than  the  central  forces  do  upon  the  work 
of  the  motion  of  a  planet  in  its  orbit.  They  change  the  direction 
of  the  motion  by  the  composition  of  the  forces,  but  it  requires 
no  additional  pressure  to  overcome  these  forces  at  right  angles 
to  the  path  of  motion  of  a  current  of  air. 


rdP 

EVALUATION  OF   TERM    -  /   -  155 

J       p 

/d  P  . 
—  in  terms  of 
p 

the  temperature,  eliminating  the  density.     This  can  be  done  by 
substituting  from  (62), 


p   ~  po  P  To         po  TQ        P~     T0        P' 

Substituting  and  using  the  values  of  the  constants  in  Table 
3,  omitting  henceforward  (Qi  —  Qo), 

^     ~f  '-£-£.•*?<*-«'>  +  £'  $<>-*• 

Integrating  between  limits  for  natural  logarithms, 

(425)  log  Po  -  log  P  =  ^^  (f-qf)  +  («  -  so). 

(426)  logPo-logP=I^5.^(f-?02) 


7991.04    T 
The  same  formulas  in  common  logarithms  become. 

1 


(427)    log  Po  -  log  P  = 


1321837 


(428)     log  Po  -  log  P  =  3^2  '  T° 


18400     T  '  gQ 
(429)     The  constants  are  derived  in  succession: 


574.067  =      .  287.033  = 

./  0  ^ 

156720  =  2  g0lQ.  7991.04  =  10. 

1321.837=^.  660.919  =  ^ 
TO  M  1  o 


360862  =  -.      18400  =   . 


156  THE  HYDRODYNAMICS   OF   THE  ATMOSPHERE 

The  second  term  of  (428)  is  the  same  as  in  formula  (159),  to 
which  static  barometry  is  usually  confined,  but  the  term  in  the 
velocity  should  be  added  for  accurate  computations.  In  the 
integrations  between  two  points  the  mean  temperature,  JIG  =  Tm, 
and  the  mean  gravity,  gi0  =  gm,  of  the  air  column  should  be 
employed,  as  already  explained. 

II.  Since  by  the  Boyle-Gay  Lussac  Law,  we  have: 

I  _RT 
P   "      P1 

this  value  can  be  introduced  into  the  equation  (419),  so  that 
(430)      -/  -j-  =  ~  (f  -  <?o2)  +  ^  (*  -  *o). 

This  is  correct  because  2R  =  574.067,  by  Table  3,  and 


0  79904' 

III.  Since,  by  equations  (176)  and  (178),  we  have,  for  n  =  1, 


this  value  can  be  introduced  into  the  equation  (419).     Then, 

/fi  P  i 

-    = 


/P\AzI      (T\n 
Since  (-5-)  *    =  l^r)  ,  we  find  again,  for  n  =  1, 

V-t      '  ^±     ' 


-t   0'  ^±  0 

(433)     -^-= 


In  case  w  is  not  equal  to  unity,  and  the  non-adiabatic  tem- 
perature distribution  of  the  air  is  considered,  we  have: 

(434)      -  /  ^  =  £j^  [i  (?2-?o2)  +  g  (z  -  to)  ]. 

IV.  In  case  it  is  desired  to  reduce  the  equation  of  motion  to 
a  form  where  the  standard  density  is  unity,  we  shall  have,  p0  =  1, 
and 


/dP 
157 
P 
k-\ 

(435)  —  =  P»17*  .  P~l/k  =  (R,  r0)v*.  j  "  .     Hence, 

1-fe 

(436)  -  flj-  =  (1goro)i/*  [J  (?2  -  go2)  +  *  (*  -  *o)l. 

This  form  eliminates  the  temperature  and  throws  all  the 
discussion  upon  P,  being  an  impure  form,  since  P  can  be  found 

only  by  trials. 

CdP 
V.  It  is  often  desirable  to  integrate  the  term  —  J  --  with- 

out using  the  mean  temperature  of  the  column,  and  this  can 
be  done  by  determining  the  mean  density  of  the  column  pm. 
Then, 


(437)      -=- 


P  Pm 

(Qi  -  Co),  and, 

(438)  -  Pl~P°  =  +Hl  (Cpa  -  Cp10)  (7\  -  To)  - 

Pm 

»i  C#io  (Ti  -  To). 

Numerical  Check  on  the  Two  Systems  of  Formulas 

We  have  already  found,  if  the  humidity,  gravity,  and  moun- 
tain terms  are  now  omitted,  that 

(439)  log  f  =  log  f  =  1840027067<6,,  by  (159),  and  that 

(440)  log  J°  =  log  |?  =  -  n  ^  (log  T  -  log  T0),  by  (182), 

so  that  the  two  methods  of  computation,  through  the  mean 
temperature  of  the  air  column  6  and  the  gradient  ratio  n  can  be 
compared  and  checked.  They  prove  by  trial  examples  to  be  in 
agreement. 

Numerical  Evaluations  of  the  Pressure  Gradient 

It  is  necessary  to  reduce  the  difference  on  the  barometric 
pressure,  at  any  distance    apart  on  a  horizontal  level,  to  a 


158 


THE  HYDRODYNAMICS  OF  THE  ATMOSPHERE 


standard  distance  which  is  taken  as  1  degree  =  111  111  meters 
on  the  surface  of  the  earth. 

G  =  the  barometric  difference  for  the  distance  111111 
meters. 

DQ  =  1  degree  or  111  111  meters  on  the  surface  of  the  earth. 

A  =  any  distance  on  the  surface. 


(441)     £  = 


D0         111111 
DC 


dB 
dx 


A 


?!    ~    B 

A 


.    Hence, 


(442)      G  =  ^  (ft  -  5). 


Lower  pressure 


Higher  pressure 


B   dx 


FIG.  17.    The  derived  gradient  for  the  distance  of  I  degree. 

The  term  —  —  —  can  be  evaluated  in  many  ways. 
Since  P  =  g0  Pm  B  =  go  p,  we  have: 


fAA»\ 

(443)     —  =    goPm-^=S<>2-x- 


(444) 


=  1.200  G  (meter)  = 

0.0012  G  (mm.). 


dp 

dx 


£ 

111  111 


(445)     5f  =  Pm-     —  =  0.12236  G  (meter) 


0.00012236  G  (mm.). 
Hence,  from  these  formulas  and  others  preceding, 


(446)  -7t^  =  -7F 

P    U  *v  Po   -£  0 


dP 


=  _  gQ 


'o    dx 

go  pmBnTd  log  P 

Po        TO     dx 


NUMERICAL  EVALUATIONS   OF  PRESSURE   GRADIENT        159 


7  1£  =  -  7  ^"  H  =  ~  7  Lao°  G  (meter)  - 

-  -  0.0012  G  (mm.). 
p 

(448)  -7H=~7p-lf=-7  °-12236  G  (meters)  = 

-  -  0.000  12236  G  (mm.). 
P 

T/fe  Evaluation  of  the  Ratios  -j-^  and  -7—  . 

a  r>         ax 

We  have,  from  Fig.  17, 


(450)     -j—  =  y^  =  jr  =  jj  (for  the  top  of  the  homogeneous 

atmosphere). 

dh_  h0       _Do±          Pi       I         I       7991.04 

d  B  ~  BQ  -  B  ~  G  Di  ~  Bi  -  BDt  ~  Bt  ~     0.760 

10514.5. 

The  pressure  B  =  0  at  the  top  of  the  homogeneous  atmo- 
sphere. BI  is  the  pressure  of  the  atmosphere  0.760  m.,  and  /  = 
R  T  =  the  height  of  the  homogeneous  atmosphere  7991.04. 

For  the  ratio  -7—  ,  we  obtain,  by  (449)  and  (451), 
a  x 

dh  _d_hL   Bo-  B  __  dh^  t  G_  _  10514.5 
'     dx~dB'     D0      ~dB'D<>~niin 

0.09463  G. 

This  makes  it  possible  to  compute  approximately  the  height 
of  the  required  isobar  above  the  surface,  at  the  horizontal 
distance  from  the  place  of  1  degree  =  111  111  meters. 

To  Find  the  Difference  of  Pressure  (Bi  —  B)  at  the  Distance  A 
that  will  just  balance  the  Eastward  Velocity  v 

The  eastward  velocity  of  the  general  circulation  v  produces  a 
pressure  directed  southward  along  the  meridian,  and  it  is  required 


160  THE   HYDRODYNAMICS   OF   THE   ATMOSPHERE 

to  find  the  difference  of  pressure  in  the  meridian,  higher  to  the 
south  in  the  northern  hemisphere,  that  will  keep  the  velocity  of 
motion  directed  exactly  eastward.  By  equation  (414),  for 

steady  motion,  -77  =  0,  and  neglecting  the  term  in  w,  we  have 
for  v  in  meters  per  second, 

(453)  —  ;—  =  cos  6  (2  co3  +v  )  v. 
p  d  oo 

Since  d  P  =  g  p  d  k,  this  becomes: 

(454)  g  -T-  =  cos  d  (2  co3  +  v)  v. 

d  oc 

By  equations  (452)  and  (442), 

f      .        10514.5    D0(Bi-B)  ,0  N 

(455)  g  —  —  .  -    -~     -  =  cose  (2  co3  +  v)  v.    Hence, 


(456)    Bi-B 

For  any  temperature  and  pressure  other  than  the  standard, 
T0  =  273  and  B0  =  760  mm.,  if  we  take  A  =  DQ  =  111  111  111, 
so  that  BI  =  BQ,  that  is,  DI  =  the  pressure  at  a  distance  of 
1  degree  southward  on  the  meridian,  we  have,  for  g  =  go, 


B"  "  B  =  •      •  Wo  •  cos  e  (2 


The  angular  velocity  of  the  rotating  earth  is  2  co3  =  0.0001458 
and  if  v  is  neglected,  we  find, 

T> 

(458)     BQ  -  B  =  0.05644  ^  v  cos  0  (in  millimeters). 

It  should  be  learned  from  this  example  how  to  apply  the 
general  equations  of  motion  on  the  rotating  earth  to  special 
cases,  by  making  the  proper  limitations  in  the  use  of  the  terms. 

The  Angular  Velocity  of  the  Earth's  Rotation,  a>3 

Allowing  for  the  sidereal  time  the  angular  velocity  is, 
o  2  V  ^  141  'iQ 


{(23X60)  +56}  60  660 


LINEAR,    ABSOLUTE    AND   RELATIVE    VELOCITIES  161 

(460)  2  co3  =  0.000145846. 

The  Linear,  Absolute  and  Relative  Velocities 

The  absolute  linear  velocity  at  any  latitude  is 

(461)  v'  =  r  o>3  sin  0 

at  the  distance  r  =  R  -f-  h  from  the  center  of  the  earth,  where 
R  =  6370191  meters  of  208996600  feet,  and  h  is  the  height  in 
the  atmosphere  above  the  surface.  The  relative  linear  velocity 
of  a  body  moving  eastward  over  the  surface  of  the  earth  is 

(462)  v  =  r  v  sin  0.     Hence, 

,Aao.  v'  2vf  cos  0      2v'cot0 

(463)  co3  =  —  r—  -  ,  and  2  co3  cos  0  = 

' 


,._.* 

(464) 

(465) 


, 
r  sin  0  '  r  sin  0 

z>  cos  0      v  cot 


It  follows  that  formulas  (414)  can  be  written  in  another 
form,  remembering  that  co  =  n  for  the  angular  velocity, 

18P      du      cot  0  ,  «  w 

(466)     ----  =  —---    r  (2  9*4-  9)  v  +  ->  r  . 

3^       f/  ^          r  r 


rf  z;       cot  0  ,  .w 

ay= 

9P 


In  some  respects  the  fourth  form  of  polar  equations  offers 
distinct  advantages  in  practical  computations,  especially  by  the 
aid  of  a  few  auxiliary  tables,  such  as  Tables  104,  105,  106  of  the 
Cloud  Report. 


162       THE  HYDRODYNAMICS  OF  THE  ATMOSPHERE 

Evaluation  of  the  Barometric  Gradients  in  the  Fourth  Form  of  the 
Polar  Equations  of  Motion 

From  equations  (446),  (5),  and  (449),  we  have: 

1  8P        1  BoT  Gx 

(^7)    7  9^  ==  PO  ~B  fi  '  *°  Pm  '  Do" 
,      ,     _!_  8P  _        760  X  9.806  X  13595.8         T 
1      '      P  9#   "  1.29305  X  273  X  111  111  111    B    x  = 

0.0025833  ^Gx. 

£> 

d  u 

For  steady  motion,  -7-7=0,  and  neglecting  the  term  in  w, 
a  t 

in  (466) 

(469)  -  0.0025833  ^  Gx  =  -   ^^  (2  v'  +  »)  »•    Hence, 

r>  K 

(470)  G,  =  387.1  1  •  ~  (2  ^  +  »  )  ». 


Gz  =  387.1  (2  «'+»)r  +       -  *. 

The  gradients  are,  in  millimeters  of  mercury, 
Gx  =  (Bi  —  B)X)  along  the  meridian, 
Gy  =  (Bi  -  B)y,  along  the  parallel  of  latitude, 
Gz  =  (Bi  -  B\  =  along  the  vertical. 
As  an  example  of  the  computation  take, 

The  north  polar  distance,  0    =  90  -  0  =  30°, 
The  radius  of  the  earth,  R  =  6370191,  B  =  700  mm.,  T  = 
260°  C. 

The  angular  velocity,  2  co3      =  0.00014584, 
-    The  eastward  velocity  of  the  earth  at  d,  vr  =  Rn  sin  0, 

The  relative  eastward  velocity  over  the  earth,  v  =  40  meters 
per  sec. 


EVALUATION   OF   BAROMETRIC    GRADIENTS 


163 


Logarithms 

By  Tables 

104  and  106 

387.1 

2.58782 

v' 
^  cot  0.2v 

0.005052 

700 

2.84510 

cot  0  .  v  v 

0.000435 

cot  30° 

0.23856 

0.005487 

(2  i/  +v)v  4.30492 

Logarithms 

9.97640 

cot0.v  [-j£  4- 

»)          7.73933 

378.1 

2.58782 

r 

2.41497 

B 

2.84510 

R 

6.80415 

3.17225 

0.75728 

9.21912 

T 

2.41497 

5.717 

0.75728 

Logarithms 

R  =  6370191  6.80415 

2  u>3  =  0.000145846.16388-10 
sin  0  =  sin  30°  9.69897 

2v'  =  464.5  2.66700 

v  =  40.0 

2  v'  +  v  =  504 . 5 
(2vf   +  v)v  =  2018.0 

Cx  =  (Bi  -  B)x  =  5.717 


Application  of  the  General  Equations  of  Motion  to  the  Local  Circula- 
tions in  the  Earth's  Atmosphere 

The  Local  Circulations 

The  circulations  of  the  earth's  atmosphere  can  be  con- 
veniently analyzed  under  two  classes:  the  first,  or  general 
circulation,  including  the  large  movements  that  are  primarily 
related  to  the  axis  of  the  earth  as  the  line  of  reference  for  the 
angular  motion,  and  the  second,  or  local  circulation,  including 
the  minor  movements  that  are  referred  to  axes  which  are  wander- 
ing over  the  surface  of  the  earth.  The  general  circulation  takes 
account  of  the  great  polar  whirls  covering  the  entire  hemisphere, 
one  north  and  one  south  of  the  equator,  including  the  trade 
winds  in  the  tropics,  the  eastward  drifts  in  the  temperate  zones, 
and  the  minor  circulations  near  the  poles.  These  great  zonal 
currents  break  up  into  localized  circulations,  as  determined  by 
the  ocean  and  land  areas,  which  constitute  the  first  disintegration 
of  the  general  circulation  into  smaller  circulations.  The  true 
local  circulations  are  commonly  known  as  cyclones,  anticyclones, 
hurricanes,  tornadoes,  and  waterspouts,  and  these  are  referred 
to  axes  which  move  over  the  earth  in  paths  that  are  determined 
by  their  relation  to  the  breaks  in  the  normal  general  circulation. 
Finally,  there  are  very  numerous  minor  whirls,  as  eddies,  small 
vortices  of  many  types,  which  constitute  the  effectual  internal 
friction  through  the  operation  of  the  law  of  inertia  in  the  moving 
masses.  The  analysis  of  these  motions,  by  the  application  of 
the  laws  of  hydrodynamics  and  thermodynamics  given  in  this 
chapter,  determines  the  principal  problems  in  theoretical  meteor- 


164  THE   HYDRODYNAMICS   OF   THE   ATMOSPHERE 

ology.  It  is  proposed  to  set  forth  the  main  features  of  this 
subject  with  sufficient  fulness  to  guide  other  students  to  the 
problems  of  the  research  which  are  pressing  for  solution. 

Discussion  of  the  Cylindrical  Equations  of  Motion 

The  cylindrical  equations  of  motion  are  most  convenient 
for  application  to  the  discussion  of  the  phenomena  of  the  local 
circulations  in  cyclones,  hurricanes,  and  tornadoes.  There 
have  been  several  attempts  to  adapt  these  equations  to  the 
observed  data,  and  the  two  best-known  systems,  that  of  Ferrel 
and  that  of  the  German  School  of  Meteorologists,  will  be  briefly 
mentioned  before  taking  up  the  form  of  vortex  that  I  have  been 
led  to  adopt  in  my  researches.  It  will  be  desirable,  at  the  outset, 
to  assume  that  the  motion  is  symmetrical  about  the  z-axis  in 
cylindrical  co-ordinates,  and  that  the  isobars  are  centered  as 
circles  upon  this  axis,  though  this  is  not  the  case  in  nature, 
except  for  waterspouts,  tornadoes,  and  hurricanes.  The  pure 
vortex  law  does  not  apply  directly  to  cyclones  and  anticyclones, 
and  the  disturbing  terms  which  make  the  transition  between  pure 
and  impure  vortices  can  be  studied  only  by  comparing  the  pure 
vortex  underlying  a  cyclone  with  the  data  obtained  by  observa- 
tion. It  has  been  difficult  for  meteorologists  to  do  this,  because 
the  actual  conditions  in  the  free  air  above  the  surface  are  found 
only  indirectly  by  computation,  or  in  an  inadequate  manner  by 
occasional  ascensions  with  kites  and  balloons.  In  recent  years, 
however,  enough  data  have  been  accumulated  to  make  it  possible 
to  advance  these  studies  in  the  right  direction.  It  will  greatly 
assist  those  who  are  engaged  in  the  study  of  the  atmosphere 
above  the  ground,  if,  in  planning  and  executing  the  observations, 
the  fundamental  principles  upon  which  the  actual  motions  must 
depend  are  clearly  understood. 

Ferret's  Local  Cyclone 

If  the  vortex  is  assumed  to  be  symmetrical  about  the  z-axis, 
and  the  friction  terms  k  u,  k  v  can  be  neglected,  the  second 


FERREL'S  LOCAL  CYCLONE  165 

equation  in  the  second  form  of  cylindrical  equations  (406)  be- 
comes: 


(471)     -      +  (2  co3  cos  6  +  v)  u  =  0. 

w 
-7- 
d  t 


v  dw 

Substituting,  v  =  —  ,  u  =  -7-,  and  multiplying  by  tcr,  this 
" 


becomes, 

ozo'  d  v 

(472)  2  co3  cos  0.    zrr  -7-  +  tzr  -y-  +  »  —  =  0. 

Integrating,  we  have  for  each  particle  in  gyration: 

(473)  ttr2  co3  cos  6  +  ttr  y  =  -or2  (o>3  cos  0  +  v)  =  c. 


fed 
initial  value  of  v, 


m 
(474)     Take  C  =  - —  —  for  the  entire  gyrating  mass,  if  Vo  is  the 


f  or2(w3  cos  0  +  v)  dm 

(475)  C  =  -  -  =  I  w02  (co3  cos  0  +  Vo). 

7W 

If  the  initial  gyration  is  zero,  v0  =  0,  and  we  have: 

(476)  -or2  (  co3  cos  0  +  ~    =  ^o2  co3  cos  ^. 


This  is  equal  to  the  moment  of  inertia  of  the  whole  mass  at 
the  distance  \  tffo.    We  obtain: 

(477)     —  =     ~^-2  o)3  cos  6  —  co3  cos  0,  and 


/  7K     2  \ 

v    =  \^~~2  —  l)  ^co3cos0,  the  tangential  velocity  at  to-  0. 

If  the  tangential  velocity  vanishes,  v  =  0,  for  co  =  R,  we  have: 

(478)     R2  =  ^,  and  R  =  0.707 


If  a  cylinder  with  the  radius  W0  is  drawn  around  the  gyrating 
mass,  then  at  the  distance  0.707  ttr0  the  velocity  is  zero;  inside 
this  R  the  fluid  rotates  in  one  direction  and  outside  of  it  the 
rotation  is  in  the  opposite  direction.  The  general  circulation  of 


166  THE   HYDRODYNAMICS   OF   THE   ATMOSPHERE 

the  atmosphere  is  arranged  to  operate  in  this  way,  the  air  north 
of  the  latitude  35°  16'  rotating  eastward,  and  the  air  south 
of  this  parallel  rotating  westward.  If  water  be  enclosed  in  a 
cylindrical  vessel,  and  an  upward  current  be  formed  in  the  center, 
by  heating  the  lower  surface,  or  by  a  wheel  operating  in  the 
water  to  raise  a  central  column,  it  will  circulate  in  this  way. 
The  water  will  gyrate  about  the  central  axis  as  it  rises,  till  at 
the  distance  0.707  or0  from  the  center  the  gyration  ceases,  while 
beyond  this  radius  the  gyration  reverses  and  the  water  descends 
in  a  ring  bounded  on  the  outside  by  the  vessel.  Ferrel  supposed 
that  this  type  of  vortex  would  represent  the  local  cyclone  as  well 
as  the  general  circulation,  but  this  is  not  the  case.  In  the 
general  circulation  there  is  a  fixed  mass  in  gyration,  as  there  is 
in  the  cylindrical  vessel  of  the  experiment,  but  observations  show 
that  the  cyclone,  hurricane,  and  tornado  are  constructed  by 
means  of  another  type  of  vortex  in  which  the  mass  is  continuously 
changing.  Other  mechanical  difficulties  are  mentioned  in  the 
Cloud  Report,  1898. 

The  German  Local  Cyclone 

In  the  second  form  of  cylindrical  equations  (406),  the  second 
equation  becomes: 

(479)  fi  + ** +  **.+-**--(*, 

where  /I  =  2  co3  cos  9  and  k  is  the  coefficient  of  friction. 
This  equation  has  two  solutions,  and  the  vortex  has  been  divided 
into  an  inner  and  an  outer  part  to  correspond  with  them.  Thus 

First  Solution  Second  Solution 

(Inner  Part)  (Outer  Part) 

(480)  Radial  velocity  «  =  -  -«  or.  u  =  —  — . 

—  "ZET 

A          c  *    c 

(481)  Tangential  velocity     v  =  7—    ;   •  -£U .     v  =  -r  — . 

K  —  C        A  K>     to 

(482)  Vertical  velocity         w  =  c  z.  w  =  0. 


THE   GERMAN  LOCAL  CYCLONE  167 

The  constant  c  depends  upon  the  dimensions  of  the  vortex, 
and  must  be  determined  by  observations.  These  values  of  u, 
v,  w,  can  be  readily  verified  by  substituting  in  equation  (479). 

If  we  take  the  following  current  functions,  called  the  Stokes 
functions,  and  the  vortex  law, 


(484)  w  =  +  - 

(485)  v  W  =  $ 

there  results  for  the  two  solutions, 

First  Solution  Second  Solution 

(Inner  Part)  (Outer  Part) 

(486)  Current  c 

•i-i         A  •  '/'i   =  T*   tCT^  2  \l/\   =    -f-  C  2. 

Function.  !         2 

(487)  Radial  1  8^1  c  1  9^i          c 
Velocity.  u  ~      "  -or  3z  ~     ~  ~2™'     w  "     ~ttr8lT=    ~m" 


(488)  Vertical 

iri       v  W  =    +  —  7^-=-  =    +  C  Z. 

Velocity.  -or  9w 

(489)  Vortex  I     c 
Law.                         ~~  k  —  c  2 

(490)  V 


£-c  2*  A;  0* 

It  is  seen  that  two  forms  of  the  current  function  are  required 
to  satisfy  the  general  vortex  law  which  will  be  deduced  later. 
Even  if  the  constants  k  and  c  could  be  determined  the  solution 
is  not  consistent  for  the  entire  vortex  in  either  the  inner  or  the 
outer  part.  There  is  in  nature  no  such  division  of  the  vortex, 
that  is,  there  is  no  outward  part  without  vertical  velocity,  as 
compared  with  an  inward  part  having  vertical  velocity,  w  =  +  c  z. 
It  is  for  this  reason  that  the  application  of  these  formulas  to 
the  cyclone  has  not  been  successful.  The  cyclone  is  constructed 
upon  quite  different  principles.  The  solutions  of  the  second 
equation  of  motion  can  be  satisfied  by  yet  other  values,  which 


168  THE   HYDRODYNAMICS    OF   THE   ATMOSPHERE 

give  a  consistent  current  function  for  all  values  of  the  currents 
and  the  vortex  law. 


The  General  Equation  of  Cylindrical  Vortices 

If  the  definition  of  a  cylinder  be  extended  to  include  any 
figure  of  revolution  formed  by  rotating  any  line  as  a  generatrix 
about  an  axis,  the  vortices  of  meteorology  can  be  designated 
as  cylindrical  or  columnar  vortices,  the  axis  being  approximately 
vertical  in  direction.  The  cylindrical  equations  of  motion  will 
therefore  be  adopted,  and  they  are  transformed  in  the  following 
manner.  In  discussing  problems  in  vortex  motion,  it  is  con- 
venient to  use  the  current  function  ^,  which  is  deduced  from 
the  equation  of  continuity  (394)  : 


This  may  be  put  into  another  form, 

(493)     J-  --  («w)  +     ^  -  0. 
ttr         v 


This  is  satisfied  by  substituting  the  velocities, 

(494)  «=--p, 

ZD-  8s 

1  8^ 

(495)  *=+-^, 

which  are  known  as  Stokes's  functions. 

In  order  that  the  equation  of  continuity  may  satisfy  the 

second  equation  of  motion,  assuming  steady  motion  and  —  =  0, 

u  t 

this  becomes,  from  (40l)2,  with  no  deflecting  force  and  no  friction, 

,     „.         dv  dv       uv 

(496)  w—+w—  +--=0, 

dur         90       nr 

and  it  is  sufficient  to  make 

(497)  ,  =       , 


GENERAL  EQUATION   OF   CYLINDRICAL  VORTICES  169 

so  that  vor  =  ^  =  constant  is  the  usual  vortex  law,  or  in  its 
most  general  form,  v  ttr  =  a  .  \J/,  where  the  \l/  of  the  Stokes  func- 
tions is  made  to  cover  the  vortex  law  by  the  constant  factor  a. 
By  differentiation, 


9w      vs    w     w 

(499)     |?-^t*. 

dz        &  3  z 

Substituting  these  values  in  (496),  it  becomes: 


.j  ,t         _  t 

w2  aw  9  z  """  w»  82  y     w  aw  a  3     w3  a  3  ^  ' 


If  the  equation  (496)  is  multiplied  by  zcr,  since  ^—  =  0,  it 
can  be  written, 

(501)  uv  -f-  «ztr-—  +  wttr  —  -  -f-  ^  y  —  —    =  0.  and,  therefore, 

otir  02  oz  .      , 

in  the  form, 

•p.  c\ 

(502)  u—  (ttrz;)  +  w  —  (tcrz;)  =  0. 

This  shows  that  tffv  =  \f/  =  constant  is  a  solution  of  the 
equation  of  continuity.  Any  function  of  \l/  which  satisfies  this 
equation  will  be  a  solution  of  the  second  equation  of  motion. 
Inasmuch  as  there  are  several  such  values  of  ^  known,  it  is  only 
necessary  to  choose  the  one  which  is  in  harmony  with  the  observed 
phenomena  in  the  earth's  atmosphere  in  order  to  obtain  the 
solution  of  the  motions  found  in  cyclones,  hurricanes,  and 
tornadoes,  or  waterspouts.  Hence  an  arbitrary  function  of  ^, 

(503)  wz>  =/(*), 

is  a  solution  of  the  second  equation  of  motion. 

The  potential  and  the  pressure  terms  in  the  first  and  third 
equations  can  be  eliminated  by  the  following  process.  From 
(401)  in  the  case  of  symmetry  about  the  2-axis, 

8V        dP       du  .      du          du       v* 

-- 


8V        8P        dw          dw          dw 


170  THE   HYDRODYNAMICS   OF   THE   ATMOSPHERE 

Differentiate  the  first  equation  to  3  z,  the  second  to  3  w, 
and  subtract  with  the  result, 


,      v      __  _  _ 

=  at  \d~z  ~  dv)  +  d^     z     9 


3  w\         3    /  fl2  \ 
ary 


dm         dz 
We  derive  the  following  auxiliary  differentiations: 


_-     _ 

-  "     "     - 


_         _        JL 

82  "9w"     "  w 


vz 
Since  v  w  =  /  (\l/),    v2  w2  =  [f  (^)]2,     —  =  L>/  vy/J  ?  we  have, 

W  "*" 


dz  zcr  w° 

Making  these  substitutions  in  (505),  we  obtain, 


+  '          T 
^     _9 

2fU)  9/^ 

ia^.a2^\~i  az 

vaw2 "' 


Any  function  of  ^  satisfying  this  equation  is  capable  of 
giving  a  vortex  motion.  In  the  application  to  the  atmosphere 
some  simple  forms  will  be  considered  and  illustrated  by  examples. 
The  first  form, 

(511)  t  =  Aw2z, 

gives  a  funnel-shaped  vortex,  and  the  second  form, 

(512)  ^  =  A  w2  sin  a  z, 

gives  a  dumb-bell-shaped  vortex.    These  are  the  common  ones 


GENERAL  EQUATION   OF   CYLINDRICAL   VORTICES  171 

in  the  atmosphere,  as  will  be  shown  by  the  observations.  Un- 
fortunately the  motions  under  the  complex  local  forces  that 
generate  storms  do  not  often  produce  pure  vortex  motion,  but  it  is 
the  province  of  meteorology  to  consider  the  perturbations  as 
observed  and  to  give  an  account  of  their  causes. 

The  Angular  Velocity 
By  formula  (371),  the  angular  velocity  is, 


and  if  an  arbitrary  function  of  ^  is  taken, 


it  follows  that, 


By  differentiations  it  follows  that 

f^R\      8 
(516)     — 

- 


Pressure 
Since  by  formula  (418),  omitting  Q  and  7, 

(518)      -  -  =  i  (W2  +  ^2  +  ^2)  +  g  z> 

it  follows,  by  using  Stokes's  functions,  that, 

/KinN          P          X    f/8^\2    ,    /r9*\2n    ,    ia2^2 

(519)     •  7  ".«?  Ks)  +  (a"J  J  +  *  -iT 


for  one  limit.     The  difference  of  pressure  between  two  points, 
designated  by  n  and  n  +  1,  becomes,  by  using  the  mean  density 


172  THE   HYDRODYNAMICS    OF   THE   ATMOSPHERE 

(520)        P  P        .  -f-     \  -1  fY^V  +    /^ 

(520)    Pn+1-Pn-  2  -( 


_£„,      /«i\         /a\ 

2  UwJ.  ~  U2  J.+ 


• 


The  Application  of  the  Vortex  Formulas  to  the  Funnel-shaped  Tube 

Employing  formulas  (494),   (495),   (497),   (503),  we  readily 
obtain  the  following  group  of  relations, 

r  w  Current 

^      '     ^  "  "2  ™  '          function. 

(w>\    r      JL  JL         ^L  _^  Vortex 

=  Tff*z          =  vz      '  iff  2z  constant. 

1  8i£  \L>  ww       v     Radial 

( 523 )  u  =  4- =  C  tcr   =  —       =  —  =  — .       , 

-or  82;  ttrz  222     velocity. 

.      .  \1/  \1/  w  Tangential 

(524)  v  =  —          =Ciff  z  =  —        =-  —  -&  =  uz.      ,     .. 

-or  -or  2  velocity. 


(525) 

2  ttr 


Vertical  velocity. 


The  Application  of  the  Vortex  Formulas  to  the  Dumb-Bell-Shaped 

Tube 

(526)  ^  =  A  TB*  sin  az. 

/rr_N  18^  A  aw2  cos  #3 

(527)  w=  —  —  ^—  =  —  -  -  =  —  A  a  TX  cos  az. 

TX  d  z  iff 

/Coo\  a  $  A  aw2  sin  az 

(528)  v  =  —        •     =  +  -     -  =  +  A  a  iff  sin  az. 

TX  Tff 

/__  N  1  8  ^  2  A  ttr  sin 

(529)  *  = 


APPLICATION   OF   VORTEX   FORMULAS  173 

The  Total  Pressure 
For  the  funnel-shaped  tube,  omitting  the  expansion  terms, 

(530)  -  -  =  J  (u2  +  v2  +  w2)  +  gz  =  |  C2  (ttr2+  w2z2  + 

4s2)  +  g  2  -f-  constant. 

For  the  dumb-bell-shaped  tube, 

p 

(531)  --  =  ±  (A2a2  m2  cos2  az  +  ,42  a2  tcr2  sin2  az  + 

4  yl2  sin2  az)  +  gz  +  constant. 

p 

(532)  -  —  =  ±A2a2iff2+2A2  sin2  as  +  g  z  +  constant. 

p 

(533)  -  =  §  A2  a2  tcr2+  42  (1  '-  cos  2az)  +  gz  +  constant. 

It  should  be  noted  that  the  signs  of  the  Stokes  functions 
have  been  taken  opposite  to  one  another  in  the  funnel-shaped 
and  the  dumb-bell-shaped  vortices.  This  is  because  it  is  more 
convenient  to  place  the  plane  of  reference  for  the  funnel-shaped 
vortex  at  the  base  of  the  cloud  from  which  it  is  developed,  with 
the  positive  direction  of  the  z-axis  downward,  while  in  the 
dumb-bell-shaped  vortex  the  first  plane  of  reference  is  taken  at 
or  below  the  surface  of  the  sea  or  ground,  and  the  positive 
direction  of  the  z-axis  is  upward  to  the  second  plane  of  reference. 
These  will  be  explained  further  by  diagrams  and  examples. 

The  Relations  Between  Successive  Vortex  Tubes 

A  vortex  is  so  constructed  that  a  section  through  it  perpendi- 
cular to  the  2-axis  at  any  height,  z  or  az}  cuts  off  a  series  of  rings 
so  regulated  in  size  that  the  successive  radii  stand  in  a  constant 
ratio  to  each  other.  Take  this  ratio, 

(534)  P  =  ~,    and 


(535) 


174  THE  HYDRODYNAMICS   OF   THE  ATMOSPHERE 

If  tffn  is  the  radius  of  the  outer  ring  Wi,  and  or2j  or3,  zcr4  ...  of 
the  successive  rings  inward,  then, 


/KQAX  1  2  3 

(536)     p=_=_=_ 


This  constant  ratio  p  plays  a  very  important  part  in  the 
computation  of  these  vortices,  and  it  is  found  that  we  can  pass 
from  one  value  of  the  radius  and  the  velocities  to  the  w  others 
in  succession  by  employing  the  following  formulas: 

(535)  Ratio  of  the  radii.  log  p  =  log  -^-. 

Wn+l 

(536)  Vortex  constant,  log  Cn  =  log  d  +  2  n  log  p. 

(537)  Radii  of  rings,  log  wn  =  log^i  —  n  log  p. 

(538)  Radial  velocity,  log  un  =  log  HI  +  n  log  p. 

(539)  Tangential  velocity,      log  vn  =  log  Ui  +  n  log  p. 

(540)  Vertical  velocity,  log  wn  =  logwi  +  2n  log  p. 

(541)  Horizontal  angle  i,    log  tan  i  =  constant. 

(542)  Vertical  angle  >?,         log  tan  y  =  log  tan  ^  +  w  log  p. 

(543)  Time  of  one  rotation  /,     log  /„  =  log  ti  —  2  n  log  p. 

(544)  Volume  through  rings,          V  =  TT  (2ttrn  —  ttr2n+i)  wm  = 

constant. 

(545)  Centrifugal  force,    log  (-)     =  log  (-)    -f  3  n  log  p. 

\w/.n  Vw/i 


(546)  Barometric  pressure,  logEj     =  log        -  +  log 

Dn-\  —  Bn  Wn+i 

(547)  Total  velocity  g,  #  =  (w2  -f  v*  +  ^2)J  =  z;  sec  i  sec  w. 

The  relations  shown  by  these  formulas  will  be  made  clearer 
by  a  diagram  giving  the  connection  between  the  angles  and  the 
velocities. 

(548)  tani-.  (549) 
(550)     tan  y  = 


a  sin 

«tfi 

-   —.. 
v  sec^ 


RELATIONS   BETWEEN  SUCCESSIVE   TUBES 


175 


ttr,  0  =  the  cylindrical  co-ordinates  of  a  point  on  the  x  y  plane 
q  (u,  v,  w)  =  the  co-ordinates  of  the  velocity  at  the  point  (zEr,0,z), 
ff  (u}  v)  =  the  component  of  q  on  the  horizontal  plane;  i  =  the 


FIG.  1  8.    The  relations  of  the  angles  and  velocities  in  the  formulas. 

angle  from  v  to  <r,  positive  outward  from  the  tangent;  y  =  the 
angle  from  <r  to  q,  positive  upwards. 

The  Second  Form  of  the  Cylindrical  Equations  of  Motion  (406)  in 

Terms  of  the  Current  Function  \f/ 

The  equations  of  motion  for  cylindrical  vortices  can  be  readily 
transformed  into  terms  depending  upon  the  current  function  ^. 
Writing  equations  (406)  in  their  full  forms,  they  become, 

,  fn. 
(552) 


du  ,      du  ,       du      v2 
—  +  u  -  —  h  w^-  ---  2  o>3  cos  6  .  v  +  k  u. 
pottr      ot          out         oz      iff 


1  dP       dv 


1  dP      dw 


du> 


dv     uv 


dw 


It  is  only  necessary  to  develop  the  differential  terms  from 
the  velocities  given  in  (523)-(525)  and  (527)-529). 


(553) 


For  the  Funnel-Shaped  Vortex 
u  =  C  iff.  v  =  C  iff  z.  w 


-  2Cz. 


dv 

—  =  C  z. 


— 
dz 


176  THE   HYDRODYNAMICS    OF   THE   ATMOSPHERE 


8w 

8w 

w2 

u  —  =  C2Tff  . 

O'Ztf 

w  —  =  0. 
82 

-    =  C2  W  22. 

-or 

u  1) 

8y 

«z> 

ufa=0ws 

.     w;  ;r-    =  —  2  C2  ttr 
82 

z.  —  =  C2  or  3. 
-or 

dw 
u—  =  0. 

8w 
w  —  =  4  C2  2. 

(554)          «  — 


Hence  the  second  form  of  the  cylindrical  equations  of  motion 
becomes,  for  the  funnel-shaped  vortex, 

(555)     -  -        =         +  C2  w-  C2  z2  -os  -  2  tcr3  cos  0  .  v  +  k  u. 


8P 


^~  ~~  '         ==    ~~^  ~i~  4  O    2     "T*  ^  r~  k  IV . 

p  u  Z         9  / 

Multiply  the  equations  (555),  respectively,  by  8 -or, -or  8$, 
and  82,  and  integrate  for  the  total  pressure,  and  we  obtain, 
omitting  the  friction  terms, 

(556)     -   f—  = =  J  (w2+z;2+w2)-f  J  C2ttr2-  J  C222^2+ 

«/     p  p 

Substituting  the  values  of  u\  v2i,  rwi\  this  becomes : 

p 

(557) =  C2  or,2  +  4  C2  2  2+  g  2,  at  the  point  (ttr,  2). 

The  difference   of  pressure  between  two  points  (&i  z)n   and 
w+1  may  be  expressed, 

rt 

=  (C2  w2)n  +1  -  (C2  w2)n  +  4  [(C2  22)n  +i-  (C2  s2)J  + 


Pm 


It  has  been  customary  in  meteorology  to  use  the  formula 
(530)  as  an  expression  for  the  total  pressure  integral,  but  it  is 
evident  that  (556)  and  (557)  are  the  complete  forms  for  the 
funnel-shaped  vortex.  If  the  inertia  terms  are  omitted  the 
formula  becomes,  without  friction, 

(558)     --  =  |C2*r2-K2*2^2 


FOR   THE   FUNNEL-SHAPED   VORTEX 


177 


Hence,  we  can  summarize  the  result  for : 

p 

(556)  includes  the  inertia  and  the  expansion  terms;  (519) 
contains  the  inertia,  but  omits  the  expansion;  (558)  omits  the 
inertia  but  contains  the  expansion. 


For  the  Dumb-Bell-Shaped  Vortex 

From  the  equations  (527)-(529)  we  have  by  differentiation  and 
substitution,  using  \J/  =  A  ttr2  sin  az, 
u  =  —  A  a  or  cos  az. 


(559) 


(560) 


—  =  —  A  a  cos  az. 

du 

—  =  A  a2  w  sin  az. 
oz 


v    =  A  a  -or  sin  az. 
—  =  A  a  sin  az. 


oz 


cosaz. 


w  =  z  A  sin  az. 


dw 

—  =  2  A  a  cos  az. 

o  z 


u  ~  =  Az  aztff  cos2  az. 


dv 
u—  =  —A 

dw 


;^-  =  2  A2  a?  iff  sin2  oz. 

O£ 

^- 
oz 

dw 

-^-  =4A2asm  az  cos  az. 

dz 


=  —  A2  a2  -or  sin2 

•or 


wz  .,   2 

—  =  —A  a  W  sm  az  cos  <zz. 

•or 


Substitute  these  values  in  the  general  equations  (552) : 
1    dP 


(561) 


otcr 


=  ^-.  -f  A2  a  2tff  —  2  -or3  cos  0  .  v 


w. 


1  9P        3w 

—  -  —     =  ^—  +  2^42.  2  sin  a  z  cos  a  z.  a  +  F  +  kw. 
p  o  z         o  t 


178  THE   HYDRODYNAMICS    OF    THE   ATMOSPEEfcE 

Multiply  by  3  ztr,  iff  3  <£,  3  z,  respectively,  omit  the  &-terms, 
and  integrate  for  the  total  pressure,  remembering  that, 

(562)  2J  sin  az  cos  a  8  az  =  sin2  az, 

(563)  -  -  =  \(tf  +  i;2  +  w2)  +  ^2  a2  to-2  +  2,42  sin2  az  +  g  z. 

p 

The  term  for  the  velocity  square  is, 

(564)  i  (w2  +  ^  +  w2)  =  |  ^2  a2  w2  +  2  ^2  sin2  az, 

so  that  the  total  pressure  with  the  inertia  and  expansion  becomes: 

p 
(565) =  A2  a2  iff*  +  4  A2  sin2  az  +  gz,  for  the  point  (of  z). 

p 

The  total  pressure  without  the  expansion  is: 

(566)  -  -  =  ±A2 a?  w2  +  2  A2 sin2 az  +  gz. 

p 

The  total  pressure  without  the  inertia  is : 

(567)  --  =  |^2a2^2  +  2A2sm2az  +  gz. 

It  is  apparent  that  in  the  dumb-bell-shaped  vortex  the  pressure 
p   p 

difference  — — ,  required  to  overcome  the  inertia  resist- 

Pm 

ance,  is  the  same  as  that  which  is  needed  to  overcome  the  resist- 
ance to  expansion. 


The  Deflecting  Force 

The  terms  in  0,  the  polar  distance,  —  2  w3  cos  0 .  v  and 
+  2  co3  cos  6  .  u,  disappear  from  the  equation  of  total  pressure 
in  the  summation  because  we  have, 

(568)  vdiff  =  urn  d(f>, 

just  as  in  the  rectangular  co-ordinates, 

(569)  v  d  x  =  u  d  y, 

which  shows  that  the  deflecting  force  is  at  right  angles  to  the 

ifi   uv 
direction  of  motion.     The  centrifugal  force  — ,  — ,  being  at  right 

angles  to  the  direction  of  motion  and  induced  by  the  velocities 
w,  vy  together  with  the  deflecting  force,  has  no  influence  upon 


THE   DEFLECTING   FORCE  179 

the  circulation  except  to  change  the  direction  without  producing 
acceleration.  In  the  same  way  a  planet  falls  toward  a  body  ex- 
erting central  force,  and  thus  moves  in  an  orbit  about  it,  but 
the  velocity  in  the  orbit  is  not  changed  by  these  forces  acting  at 
right  angles  to  the  direction  of  motion. 

The  Force  of  Friction 

The  viscous  friction  in  the  atmosphere  is  a  very  small  quan- 
tity, and  k  would  be  a  small  coefficient  were  it  not  that  in  all 
large  movements  of  the  air  there  are  numerous  small  vortices 
produced  within  and  carried  along  in  the  great  current.  These 
minor  whirls  have  a  strong  force  of  resistance  and  they  are 
largely  concerned  in  frittering  down  the  energy  of  motion 
contained  in  the  large  current.  It  is  customary  to  take  the 
term  expressing  friction  proportional  to  the  velocity,  k  u,  k  v,  k  w. 

This  is  a  subject  that  has  not  been  satisfactorily  cleared  up, 
and  it  will  require  much  careful  research.  There  can  be  no  doubt 
that  k  is  a  variable  coefficient,  and  differs  widely  in  tornadoes 
passing  over  a  city  or  rough  country  from  that  in  a  cyclone 
over  an  ocean  area.  It  is  not  certain  that  the  velocity  enters 
the  equations  as  the  simple  first  power,  but  that  remains  to  be 
determined.  The  resistance  due  to  the  friction,  whatever  function 
may  be  found  to  express  it,  acts  along  the  line  of  motion  to 
retard  the  velocity,  so  that  the  pressure  difference  must  increase 
to  overcome  this  type  of  resistance. 

We  may  write  the  final  equation  for  all  the  terms,  when 
the  pressure-difference  between  two  points  is  required,  using 
(vt  z)  n  +  i  and  (w,  z)n, 

p    _n+i  n+i  n 

(570)--  =^2a2^2          +  4^2sin20z 

Pm  J  n  J  n  _J  n 


The  mean  density  pm  along  the  path  between  the  two  points 
must  be  used,  and  in  general  the  mean  conditions  of  all  the 
terms  along  the  path  of  the  integration  must  be  carefully  con- 
sidered. 


180       THE  HYDRODYNAMICS  OF  THE  ATMOSPHERE 

The  Transformation  of  Energy  in  the  Circulation  of  the  Atmosphere 

The  circulation  of  the  atmosphere  is  the  process  of  the  trans- 
formation of  energy,  the  transportation  of  warm  and  cold  masses 
of  air  from  one  place  to  another  being  the  evidence  that  a  dis- 
turbed thermal  condition  is  seeking  its  normal  equilibrium. 
These  currents  are  so  complex  that  at  present  there  is  no  possi- 
bility of  working  out  a  comprehensive  system  of  equilibrium. 
The  direction  and  velocity  of  the  currents  in  all  levels  and  in 
all  latitudes  and  the  temperatures  of  the  masses,  must  be  worked 
out  by  numerous  observations  before  that  can  be  undertaken. 
All  the  integrations  heretofore  proposed  assume  that  a  nearly 
perfect  vortex  law  can  be  laid  at  the  base  of  the  discussion  of  the 
general  and  the  local  circulations,  but  as  the  vortices  on  the 
hemisphere  and  in  the  cyclones  are  very  imperfect  a  more  com- 
plicated treatment  is  necessary.  At  present  it  is  possible  to 
lay  down  only  some  isolated,  detached  propositions  which  con- 
tribute to  the  ultimate  solution  of  the  problems  of  atmospheric 
circulation.  The  following  discussions  merely  introduce  a 
subject  of  great  value,  which  is  capable  of  unlimited  development. 

CASE  I.  The  Change  of  Position  of  the  Layers  in  a  Column  of  Air 

When  a  layer  of  air  in  a  column  is  not  at  the  temperature 
which  belongs  to  its  elevation  it  must  move  upward  or  down- 
ward in  order  to  gain  a  position  of  equilibrium,  upward  if  too 
warm,  and  downward  if  too  cold  for  its  place.  This  occurs  when 
a  cold  sheet  overruns  a  warm  layer,  when  there  will  be  an  inter- 
change of  position  in  certain  streams,  which  may  have  a  vortical 
structure  more  or  less  fully  developed.  The  following  proposi- 
tions take  no  account  of  the  form  of  the  current  lines,  but  they 
explain  the  amount  of  energy  that  can  be  transformed  into  a 
velocity  q.  The  chief  imperfection  in  these  propositions  consists 
in  the  omission  of  the  powerful  heat  terms  (Q\  —  QQ). 

From  the  equations  (196)  to  (199),  we  find, 

(571)     -  — -  =  g  fe- 
Pio 


THE   CHANGE   OF  POSITION  OF   LAYERS 


181 


so  that  the  velocity  equation  becomes,  for  the  mass  M, 

(572)    J  (q? - ?02) M  =  -g(Zl-Zo)M- (»! - n0) Cpu (7\ - T0) M 

M-(Q1-Q0)M. 

The  evaluation  of  the  term  —  (HI  —  nQ)  Cpi0  (Ti  —  TO)  m 
is  difficult,  because  the  moment  a  mass  of  air  moves  up  or  down, 
it  at  the  first  instant  has  an  adiabatic  gradient,  w0  =  1,  of  ex- 
pansion or  contraction,  which  sets  up  a  minor  circulation  within 
the  mass  whose  gradient  is  HI,  so  that  this  internal  circulation 
cannot  be  followed,  and  it  is  necessary  to  treat  it  as  a  resultant 
mass  whose  general  gradient  is  n\.  We,  therefore,  omit  this 


clM2 


pfc 


fnitial  Final 

FIG.  19.    Change  of  position  of  the  layers  in  a  column  of  air. 

term,  also  the  initial  velocity  g02  and  the  initial  height  z0  for 
convenience,  and  have  for  the  kinetic  energy,  for  several  masses, 

(573)     I  m  f  —  2  [— n\  Cpio  (Ti  —  T0)  m  —g  z  m]. 

These  terms  must  be  evaluated  and  substituted  in  the  general 
formula.     (Compare  Margules'  "Energie  der  Stiirme.") 


Change  of  Position  of  the  Layers  in  a  Column  of  Air 

Suppose  that  the  thin  layer  mi  at  the  height  Zi,  pressure  Pi, 
temperature  Ti,  is  too  warm  for  its  place,  but  that  it  must  rise 
to  the  height  02  to  be  in  equilibrium,  while  a  column  Mz  of  height 
h  falls  through  a  short  distance.  The  mass  Mo  is  not  affected 


182 


THE  HYDRODYNAMICS  OF  THE  ATMOSPHERE 


while  the  mass  Mh  above  z2  falls  as  in  a  piston  without  changing 
the  pressure  or  temperature.  The  changes  in  the  mass  M2 

must  be  integrated  through  the  layers,  M  2  =   /  d  M2.    In  the 

•/o 

exchanges  the  pressure  of  mi  changes  from  PI  to  PI,  and  the 
temperature  from  TI  to  7Y;  the  pressure  of  the  differential 
layer  d  M2  changes  from  P2  to  P2',  and  the  temperature  from  T2 
to  Tz,  while  the  height  h  changes  slightly  as  the  large  mass  con- 
tracts in  falling. 


Layer 
(574) 


Initial 


Pi'  = 


Final 
g  mi  •      2Y  = 


To  evaluate  T2  we  have 

,»/     fc-l 

/r~r\  rp   ,  rp       /•*•    2     \— IT  rp       I    +        .    6_lllf  A  n  k  T 

(575J     12    «/i(-«"l*"    =  ^2  I  1 +^—   WA!  =  ^2 


_ 

"  l2 


R 


wyfe 
Substitute  these  values  in  equation  (573). 

/Vi-r/^N  1        .     ^2  *     {~*  J\ 

\oi\)j      ~2  mi  c[    =  ni  ^P] 


nCpP2' 


(577)     |  wi  g2  =  %  Cpw 
since  in  the  d  M2-term, 

(578)   m  CAO  (r,  -  r2  -/g 


)"»*  J  wi  -  g  (2  - 


f  dM2  r 

l   --  =-  gmi)  dz=-gmih. 
«/      p2  * 


The  two  gravity  terms  in  (576)  nearly  disappear  by  the 
summation.  The  available  kinetic  energy  ^  mi  q2  caused  by 
displacing  a  thin  layer  by  a  thick  layer  can  be  computed  in  this 
way,  but  there  is  no  account  given  of  the  form  of  the  currents 
produced  by  the  transformation,  nor  of  the  energy  lost  in  the 


EVALUATION   IN   TEMPERATURE   CHANGES  183 

small   internal   vortices   with    the   accompanying    inertia   and 
friction,  nor  the  energy  lost  by  radiation. 

The  Evaluation  of  JT  dm  in  Linear  Vertical  Temperature 

Changes 

Since  the  integration  of  the  term  JT  dm  may  frequently 
occur  for  a  simple  linear  vertical  gradient,  it  is  proper  to  secure 
the  general  auxiliary  theorem  that  will  express  this  term  when 
the  temperature  is  defined  by 

(579)  T  =  To  -  a  z. 

If  a  is  not  constant,  as  is  seldom  the  case  except  for  short 
vertical  distances,  then  another  solution  will  be  required. 
We  have  to  evaluate,  for  T  =  TQ  —  a  z, 

(580)  $Tdm=$TPdz. 

It  is  convenient  to  have  before  us  the  equivalents, 

P       /T\nCp      /T\—        /r\—  *  —      /T\8/Ra 


By  substitutions,  we  find, 

rz  1    f*  1    rz     t  T  \  g/Ra 

(582) 


-g/Ra     g/Ra 

T       dz. 

o 

Change  the  limits  of  integration  from  z  to  T.    Since, 

jrrt 

(583)     T  =  To  -  a  z,    dT  =  -  adz,     -  ~-  =  dz,  we  have: 


(584)         TPdz  =  --Po  T,-ga       T8ad  T  =     -  P0  T0~g/Ra 

r  jK# 

1  F  T  g/Ra  +1        Tg/Ra+l^\ 

a+ll10  ]' 


g/Ra+l 

T 


(P0T0-PT).    Hence, 


g  +  Ra 
(586)      fldm  =  CT pdz  = r — :  (P0T0-  PT), 

«/n  Jo  .  K  —  1 


t^TI 


184  THE   HYDRODYNAMICS   OF   THE   ATMOSPHERE 

The  difference  of  the  products  of  the  pressure  and  the  tem- 
perature at  two  points,  multiplied  by  the  coefficient  depending 
on  the  w-coefficient  of  the  gradient  of  the  temperature  between 
them,  is  the  integral  of  this  term.  It  is,  however,  much  simpler 
to  integrate  by  means  of  T  and  p  for  the  stratum  (zi  —  20), 

(587)  fT  dm**  fr  p  d  z  =  Tio  Pw  Oi  -  *0), 

«/0  «/0 

which  gives  close  approximate  values. 

CASE  II.  Effect  of  an  Adiabatic  Expansion  or  Contraction 
in  a  N  on-  Adiabatic  Temperature  Gradient 

Since  a  mass  in  moving  from  one  level  to  another  level  in 
the  atmosphere  begins  to  change  adiabatically,  while  the  pre- 
vailing temperature  gradient  is  non-adiabatic,  it  becomes 
desirable  to  define  the  relation  of  these  facts  to  the  velocities 
which  are  immediately  set  up  in  the  mixing  medium.  The 
equation  (573)  is  to  be  evaluated  under  adiabatic  conditions, 
by  which  it  becomes, 

(588)  i  fin  g2  =  Cp 


From       =  '  we  have'  f°r  r*  =  Tl  "  a  h' 


From  the  binomial  theorem,  we  have  the  formula, 

1 


(590)     (l-X)n=l- 


nx* 

--,  so  that 


(591)     i  m,  q*  =  Cp     ^-T,  (1  -m        +  n* .  |          - 

2 


EFFECT  OF   EXPANSION   OR  CONTRACTION  185 

(592)     1 
(593) 


The  mass  mi  is  driven  from  its  position  with  a  velocity  energy 
inversely  proportional  to  the  temperature,  so  that  warm  air  has 
less  driving  power  than  cold  air.  The  drive  depends  upon  the 
departure-ratio  n  and  vanishes  when  n  =  1,  that  is,  for  adiabatic 
expansion  in  an  adiabatic  gradient.  When  a  >  do  the  mass  m: 
is  in  unstable  equilibrium,  that  is,  too  cold  for  its  position  and 
tends  to  fall.  Example,  for  n  =  0.5,  a  =  19.747,  a0  =  9.87. 
When  a  <  aQ  the  mass  mi  is  in  stable  equilibrium.  Example,  for 
n  =  2,  a  =  4.94  <00  =  9.87.  It  is  not  possible  to  drive  the 
small  mass  mi  through  any  great  height  in  the  atmosphere, 
because  the  differential  energy  of  the  expanding  mass  sets  up 
minor  whirls  which  tend  to  interchange  the  heat  energy  by 
mechanical  effects  and  internal  friction  and  radiation.  The 

result  is  to  change  the  gradient  from  aQ  to—.    If  the  displace- 

ment of  the  mass  mi  takes  place  in  the  medium  of  gradient  a, 
then  the  drive  may  be  expressed  by  terms  of  the  form 

/r^,\     i  i     h*        /a0      aQ\       i    h* 

(594)     zmif  =  $gYimi  (-  -  -)  =  ^g^mi 

where  HI  is  the  effective  temperature  ratio  of  the  moving  mass, 
and  n  is  that  of  the  prevailing  general  temperature  gradient. 

There  are  two  primary  type  cases  of  the  distribution  of  the 
masses  of  different  temperatures:  (1)  That  in  which  they  are 
superposed,  and  (2)  that  in  which  the  masses  are  located  side 
by  side  on  the  same  levels. 

CASE  III.  The  Cher  turn  of  Deep  Strata  in  the  Column 

Let  the  pressures,  temperatures,  and  heights  be  arranged 
in  the  initial  and  final  states  as  indicated  in  Fig.  20. 


185 


THE  HYDRODYNAMICS  OF  THE  ATMOSPHERE 


The  Overturn  of  the  Deep  Strata  in  the  Column 

The  greatest  entropy  in  the  initial  state  in  1  is  less  than  the 
least  in  2,  so  that  the  cold  mass  1  will  fall  beneath  the  warm 


Tftl 


hi 


ICold 

2'  Warm 

Pi       Tfi 

p/       T. 

Pi       Ti2 

P<        T'^ 

2  Warm 

Cold 

Po         To  2 

Po        Toi 

Initial  Final 

FIG.  20.     The  overturn  of  the  deep  strata  in  the  column. 

mass  2.  The  heights  of  the  masses  will  change,  as  well  as  the 
pressures  and  temperatures.  Assume  that  P0,  r02,  h2,  Ti\,  hi 
are  known  in  the  initial  states. 

We  shall  have  the  following  formulas  for  computing  the  other 
required  terms,  in  a  non-adiabatic  atmosphere. 


T         T 
T~Tm~ 


(595)  Pi  =  F 

(596)  Ph  =  1 

Substitute  mCp(jTdm  —  jTi  d  nti)  successively,  using  (715) . 

(597)  Initial.     (7+  U)a  =  i  m  q2  =  Cp  JT  d  m  = 

^—^  (Po  T02  -  Pi  Ti2  +  P,  Ta  -  Ph  Thl)  +  const. 


(598)     Final.     (V  +  U)e  = = 

^p-j  (PO  r0i'-  P/  ra'+p/  T#-PH  TW'. 


TRANSFORMATION   OF   MASSES  187 

(599)     Kinetic  energy  =  (7  +  U)a  -  (V  +  U)e  =  \  M  q2  = 

'2     (approx.). 


6 


(600)     Heights,     h'  =  —^  (TV  -  TV). 

o 

(601)  hi  =  ^f  (TV  -  2V). 

O 

The  approximate  solution  of  this  case  (Margules)  is 
(602)     Velocity,     i  q2  =  &    fr  ?•  +~h  T 

CASE  IV.  The  Transformation  of  Two  Masses    of  Different 
Temperatures  on  the  Same  Levels  into  a  State  of  Equilibrium. 


Tft2 


FM 


1   Cold     (Sj) 

POI             Tol 

2  Warm  (S2) 
Po2             T02 

Initial 


2'  Warm 

P/ 

T/2 

l'   Cool 

T« 

Po 

To 

Final 


FIG.  21.     Transformation  of  two  masses  on  the  same  levels. 

Transformation  of  Two  Masses  on  the  Same  Levels 

Given  the  initial  data  at  the  height  h,  Thi,  TV,  PH,  the  areas 
Bi,  B2,  and  the  entropies  Si  <  S2.  Hence  by  the  formulas  we 
shall  have, 

.T    N    fife 

(603)  Pm  =  Ph(^= 

(604)  P^Ph(^= 

(605)  P;  =  PA  +  \  (PM  - 

(606)  Po'  -  P»  +  *(**«- 

(607)  r.- 


(608) 


188  THE   HYDRODYNAMICS   OF  THE  ATMOSPHERE 

(609)  Initial.     (V+lT)a  =  — ^r— r  f  (Poi  T01-  Ph  TkL  + 

O  -j  I         . 

P02  TQ2  -  Ph  Th2)  +  const. 

(610)  Final.      (V  +  U)e  =  —        \_.B  (P0;  T0'  -  P/  T,'  + 

£  i   i  _ _ 

P/TV  -  PA  rw)  +  const. 

(611)  Kinetic  energy  =  J  M  ?2  =  (F  +  Z7)a  -  (V  +  t/)e. 

(612)  Mass.         Jkf  =  -  (Po;  -  Ph). 

o 

(613)  Height.     V  =  ^  (rc'  -  zy). 

0  /^ 

(614)  k,'^(Tff'-Tu). 

An  approximate  solution  is  given  (Margules). 

(615)  Take  r  =     2~     \  T*  =  TtTt,  M  =  B  Ph-^,  = 

j.  j\.± 

B  p  h  (approx.) 

(616)  ^  M qz=  ^  M-^^-ghr. 

These  solutions  must  be  handled  cautiously  in  practice, 
because  the  internal  motions  of  the  atmosphere  introduce  ele- 
ments of  pressure,  temperature,  and  velocity  which  it  is  very 
difficult  to  follow,  and  take  account  of  in  forming  the  elements 
of  the  integrals,  and  there  is  no  term  for  the  radiation. 

CASE  V.  For  local  changes  between  two  strata  of  different  tem- 
peratures, where  on  the  boundary  the  pressure  P  =  PI'  =  P2'  and 
the  temperature  is  discontinuous 

Take  the  following  conditions: 

(617)  w2,  P2  T2,  P2'  =  P2  +  g  m1}  T2'  =  T2  {~JT^ 


(618)     mlt  Pi  T\,  Px'  =  Px  -  g.  m,,  TV  =  ^  I ^  \  ^ 


k-< 
J 


LOCAL  CHANGES  189 

The  condition  of  equilibrium  becomes,  for  PI  =  P2'  =  P, 
(619)     Kinetic  energy  =  Cp  [mi  (7\  -  TV)  +  m*  (T2  -  T2')] 

(.20)  ,c 


(621)  =  mi  fl 

r>    'T1  -|  T>    'T1 

(622)  Since    —^  =  —  and  —^  =  — ,  therefore, 

•fl  Pl  -T2  P2 

/£»OO\  1    1/f  .-,2  L»     " 

(\)Z6)  2^  M  ^     =   Wl  W2  ~  — . 

H       pi  P2 

The  kinetic  energy  inducing  an  interchange  is  proportional 
to  the  difference  of  the  densities,  and  inversely  proportional  to 
the  product  of  the  densities.  Hence,  if  strata  of  different  den- 
sities are  flowing  over  one  another  in  the  general  circulation, 
which  is  temporarily  stratified,  these  two  strata  tend  to  mix  by 
interpenetration  according  to  this  law. 

There  are  numerous  other  cases  which  can  be  worked  out  in 
accordance  with  the  law  which  may  be  assumed  for  the  distribu- 
tion of  the  temperature  in  a  vertical  and  in  a  horizontal  direction. 

Compare  "  Ueber  die  Energie  der  Stiirme,"  von  Max  Mar- 
gules.  Jahrbuch  der  k.k.  Cent.-Anst.  fur  Meteo.  u.  Erdeng. 
Wien,  1903. 

''  The  Thermodynamics  of  the  Atmosphere,"  F.  H.  Bigelow. 
Monthly  Weather  Review,  1906,  and  Bulletin  W.  B.  No.  372. 

The  General  Circulation  on  a  Hemisphere  of  the  Earth }s  Atmosphere 

While  it  is  impracticable  to  take  up  the  problems  of  the 
general  circulation  with  the  purpose  of  forming  integrals  that 
will  take  account  of  the  entire  circulation,  there  are  yet  a  few 
propositions  which  are  of  interest  in  the  premises. 

Resume  equation  (414)2,  and  limit  it  by  assuming  symmetry 

about  the  axis  of  rotation,  so  that -~—  =  0;  also,  by  omitting 


190  THE   HYDRODYNAMICS    OF    THE   ATMOSPHERE 

the  small  term  in  w,  so  that  we  shall  have  as  a  special  case, 
wherein  the  vertical  current  and  the  friction  are  omitted  from 
the  general  motion, 

(624)     cos  0  (2  co3  +  v)  u  +  -T-  =  2  cos  0  («3+v)  w  -  v  cos  0  .  w+ 


Multiply  this  equation  by  r  sin  6,  and  substitute  from  (413), 

(625^     -7-      —  V  cos  0  .  u  =  r  sin  0  —  ,  so  that, 
at  ot 

(626)  2  r  sin  0  (w3  +  v)  r  cos  0—  +  (r  sin  0)2  —  =  0. 

Integrating  for  each  gyrating  particle, 

(627)  r2  sin2  0  (co3  +  v)  =  c. 

Let  C  =  constant  for  the  entire  rotating  mass,  if  Vo  is  the 
initial  relative  angular  velocity, 

m       fr2  sin2  0  (o>3  +  V0)  d  m 

) 


(628)  c 

This  is  equal  to  the  moment  of  inertia  of  the  entire  mass  at 
the  distance  f  r.  If  the  initial  state  is  that  of  rest  on  the  rotat- 
ing earth  V  =  0.  Finally,  we  shall  have: 


(629)  r2  sin2  0  (  <o3  +  ^-J  =  f  r2,  and 

(630)  rco3sin0  +  z^  =  f>or 

(631)  v= 


This  is  the  eastward  velocity  at  the  north  polar  distance  0. 
(632)  If  v  =  0,  sin2  0  =  f  ,  sin  0  =  0.8165,  0  =  54°  44'. 

At  latitude  0  =  90  -  0  -  90  -  54°  44'=  35°  16'  the  east- 
ward velocity  vanishes  at  the  surface  of  the  earth.  Observations 
indicate  that  v  =  0  extends  upward  in  a  direction  sloping  to- 
wards the  equator  from  latitude  35°  north  and  south,  its  position 
being  indefinite  above  20,000  meters. 


TEMPERATURE    GRADIENTS   AND   VELOCITY  191 

The  equation  (624)  can  be  obtained  as  follows:  Assume  the 
vortex  principle  of  the  conservation  of  the  momenta  of  inertia, 

(633)  or2  (co3  +  v)  =  (r  sin  0)2  (o>3  +  v)  =  c    a  constant. 

This  is  not  strictly  true  in  the  atmosphere,  because  it  is  not 
circulating  in  a  perfect  vortex,  and  this  faulty  assumption  has 
been  generally  made  in  discussing  this  subject.  How  far  it 
departs  from  a  vortex  law  remains  to  be  determined  by  the 
observations.  Differentiate,  divide  by  r  sin  0  d  /,  and  we  obtain 

(634)  2  cos  6  (co3  +  v)  r-j-.  +  r  sin  B-r-.  =  0. 

d  t  (It 

Since  r-j-.   =  u,  we  find  from  (624), 

dv  dv 

(635)  v  cos  6.  »  —  Jj.+  f  $m  f  jj  -  0, 

and  this  is  the  same  as  (625). 

Ferrel  discusses  these  equations,  and  gives  some  approxi- 
mately correct  views  regarding  the  general  circulation.  Ober- 
beck's  treatment  embraces  the  three  equations  of  motion,  and 
the  solution  approaches  more  closely  to  the  flow  of  currents 
actually  observed.  The  complete  integration  of  the  system  is, 
however,  more  complex  than  has  been  admitted,  and  the  problem 
awaits  a  better  treatment.  The  actual  velocities  and  direction 
of  motion,  together  with  the  temperatures,  must  be  so  handled 
as  to  embrace  the  general  and  the  local  circulations  in  a  single 
comprehensive  solution. 

Three  Cases  of  the  Slope  of  the  Temperature  Gradients  and  the 
Resulting  Velocity  of  the  East  and  West  Circulations 

In  the  earth's  atmosphere  there  are  three  general  cases  of 
the  distribution  of  the  temperature  gradients  and  the  resulting 
circulation  which  can  be  distinguished,  though  the  solution  will 
not  be  complete  until  the  radiation  term  has  been  accounted 
for  in  the  equations  of  motion.  These  cases  are:  (l)  for  the 
eastward  drift  in  the  temperate  zone  where  the  velocity  increases 
upwards,  while  the  temperature  decreases  towards  the  pole  in  a 


192 


THE  HYDRODYNAMICS  OF  THE  ATMOSPHERE 


line  parallel  to  the  axis  of  the  earth's  rotation;  (2)  for  the  lower 
levels  of  the  westward  drift  in  the  tropics,  up  to  an  altitude  of 
about  5,000  meters,  wherein  the  westward  motion  increases  with 
the  height,  while  the  temperature  increases  towards  the  pole;  (3) 
for  the  upper  layers  of  the  westward  drift,  above  5,000  meters 
into  the  isothermal  region,  wherein  the  velocity  decreases  with 
the  height,  while  the  temperature  increases  towards  the  pole. 
The  case  (3)  seems  to  agree  with  the  conditions  observed  in  the 
atmosphere  of  the  sun,  which  has  decreasing  velocity  from  the 
equator  to  the  pole,  and  decreasing  velocity  from  the  surface 
upwards  in  all  latitudes,  accompanied  by  increasing  tempera- 
tures towards  the  pole.  The  observations  in  the  tropics  on  the 
cloud  velocities  give  an  increasing  velocity  westward  to  5,000 


ii      in 
FIG.  22.     The  relative  values  of  d  r  and  dTJf  in  three  cases. 

meters,  then  decreasing  to  11,000,  then  increasing  to  the  limit  of 

balloon  observations,  so  that  cases  (2)  and  (3)  alternate  to  some 

extent. 

or  =  the  distance  from  the  axis  of  rotation  of  the  earth. 

r  —  the  radius  from  the  center  of  the  earth. 

Draw  a  tangent  to  the  circle  at  the  initial  point  of  the  iso- 
therm. Draw  d  TX  and  d  r  to  second  points  on  the  same  isotherm, 
to  show  its  slope  relative  to  the  horizon  and  axis  of  rotation. 
We  have  to  determine  the  relation  of  the  temperatures  and  the 
velocities  of  motion  in  space  to  each  other  at  any  point  in  the 
earth's  atmosphere. 

Take   the  general  integral  of  motion   (417),   omitting   the 


TEMPERATURE    GRADIENTS   AND   VELOCITY  193 

(Q  +  J)  term,  and  supply  the  centrifugal  force  in  the  gravitation 
term,  --  |  vQ2,  where  v0  is  the  linear  velocity  of  the  rotating 
surface  at  the  given  latitude,  and  we  obtain, 

(636)  -  f—  =  |  (u*  +  vz  +  w*)  -  !  i>o2  +  g  r  +  constant. 

*/     p 

1         7?  T 

Substitute  -  =  —p-,  and  pass  to  logarithms,  also  put 

g  r  =  —  —  from  the  general  law  of  gravity. 

(637)  -  logP  .  R  T  =  |  (u*+w*}  +  i  (z;2-^2)  +  +  C. 


We  give  different  values  of  the  temperature  (7\,  T2)  to  two 
adjacent  strata  flowing  over  each  other  at  different  velocities 
(fli,  %),  but  since  the  pressures  cannot  be  discontinuous  at  the 
bounding  surface,  we  take  PI  =  P2.  Hence,  by  substitution  in 
two  strata,  and  transformation  of  the  terms  for  differentiation, 

l         l 


C, 


This  is  the  general  equation  to  be  fulfilled  at  every  point. 
Now  differentiate  (637)  to  r,  the  change  along  the  radius,  for 
two  adjacent  strata  at  pressures  PI,  P2,  and  we  have: 

^  (log  Pi-log  P2)  l         l\  (r.-7\) 


omitting  the  small  terms  in  u  and  w. 

Again  differentiate  (637)  to  -or,  holding  the  angular  momentum 
(v&)  constant  in  each  stratum.  At  the  surface  of  the  earth 
the  velocity  v<?  =  co32  tcr2.  Hence, 

d  V(?  2  co23  zcr2       2  v02 

(640)     -T—  =2co32ttr=-         -  =  --  .     Using  this  form, 
d  iff  tD"  iff 

and  differentiating  for  two  adjacent  strata, 
i  -log  A) 


_  1  /Ei2-Eo2  _  ^22-^o2\ 
-or  V     Ti  2s/ 

i  rfa2-v)r2-  (^- 

orL  ^r, 


194 


THE  HYDRODYNAMICS  OF  THE  ATMOSPHERE 


Divide  (641)  by  (639)  and  the  ratio  -r£  becomes, 
/~,«\     dv  1    r~u>i2 —  ZJo2)  T%—  (vz2 — Vo2)  '. 

(642)  ^=~^L          Tv^rT 


This  equation  connects  the  velocities  and  temperatures 
with  the  slope  of  the  isotherms,  and  it  is  capable  of  three  solutions 
which  are  expressed  as  follows: 


Case  I 


Case  II 


Case  III 


dr  _        (+) 
drf  (-) 


T2<T1 


+dr  _       (—  ) 


(+) 


T2>T1 


East 


West(lower) 


"West  (upper)  and  Sun 


FIG.  23.  The  relative  values  of  v0,  m,  v2,  Ti,  T2  in  three  Cases. 

If  dr  =  0,  and  the  isotherm  is  parallel  to  the  surface,  it 
follows  that  fa  -  v02)  T2  =  (vf  -  vf)  T!  so  that  the  crossed 
products  of  the  square  of  the  relative  velocities  at  any  point 
in  the  atmosphere  by  the  alternating  temperatures  of  the  two 
adjacent  strata  are  equal.  The  warm  stratum  assumes  greater 
velocity  than  the  cold  stratum,  in  order  to  maintain  a  gradual 
change  in  the  value  of  the  vertical  hydrostatic  pressure,  such 
as  was  developed  in  Chapter  II.  If  d  r  changes  from  0,  in  the 
three  cases  described,  and  typically  illustrated,  the  temperature 
gradients  take  on  slopes  that  respectively  balance  the  velocities 
of  the  air  movements,  generally  above  the  series  of  tangents  to 
the  horizon  in  the  tropics,  but  below  them  in  middle  latitudes, 
as  have  been  found  from  the  direct  observations  in  balloon 
ascensions.  In  Chapter  II,  it  has  been  shown  how  powerfully 
the  (Qi  —  Qo),  the  change  of  the  heat  contents  per  unit  mass 


TEMPERATURE    GRADIENTS    AND    VELOCITY  195 

from  one  level  to  another  reacts  upon  the  velocity  system,  so 
that  this  problem  cannot  be  fully  solved  through  velocity  and 
temperature  functions.  These  theorems  can  be  extended  to  very 
useful  inferences  in  the  case  of  the  sun  where  velocities  can 
be  measured,  but  where  it  is  very  difficult  to  determine  the  ab- 
solute temperatures  prevailing  in  different  strata. 


CHAPTER  IV 

Examples  of  the  Construction  of  Vortices  in  the  Earth's 
Atmosphere 

AN  extensive  computation  on  vortices  has  been  published 
in  the  Monthly  Weather  Review,  October,  1907,  and  subse- 
quent numbers,  giving  in  sufficient  detail  the  method  of  handling 
the  data.  These  comprise  the  funnel-shaped  vortex  of  the 
Cottage  City  waterspout,  August  19, 1896,  the  dumb-bell-shaped 
vortex  of  the  same  Cottage  City  waterspout,  the  truncated 
dumb-bell-shaped  vortex  of  the  St.  Louis  tornado,  May  27, 1896, 
the  De  Witte  typhoon,  August  1-3, 1901,  the  impure  dumb-bell 
vortex  in  the  ocean  cyclone,  October  11,  1905,  and  the  very 
imperfect  vortices  of  the  land  cyclones  of  the  United  States. 
In  Figs.  24,  25,  26,  are  given  typical  (w .  z)  lines  in  the  Cottage 
City  waterspout,  the  St.  Louis  tornado,  and  the  De  Witte 
typhoon,  respectively,  to  which  further  references  will  be  made. 

In  the  funnel-shaped  vortex  tube  of  the  Cottage  City  water- 
spout, the  plane  of  reference  is  at  the  base  of  the  cloud,  1,100 
meters  above  the  sea  level,  and  the  axis  extends  downwards, 
this  being  the  convenient  form  for  tornadoes  generally.  In 
the  dumb-bell-shaped  vortex  there  are  two  planes  of  reference, 
and  the  lower  one  is  placed  below  the  sea  level  while  the  upper 
one  is  placed  at  the  cloud  level.  The  axis  between  these  planes 

180 
is  divided  into  180  degrees  so  that,  a  =  1  2QQ  =  0.15,  this  being 

the  value  in  the  current  function  \f/  for  this  case.  It  has  been 
found  that  these  vortices  are  generated  at  the  cloud  base  by 
the  thermodynamic  action  of  strata  of  different  temperatures, 
and  that  they  are  propagated  downwards  to  the  sea  level  or  to 
the  surface  of  the  ground.  These  vortices  seem  to  be  cut  off 
or  truncated  at  some  distance  above  the  lower  plane  of  reference, 
and  on  this  supposition  the  vortex  laws,  when  applied  to  the 
observed  phenomena,  appear  to  meet  satisfactorily  all  the 

196 


EXAMPLES 


197 


requirements  of  the  problem.     A  few  details  of  the  computations 
will  be  introduced  in  this  connection. 


Funnel- shaped  Water  Spout 


Cloud  Level 


looVN 

200      * 

300 

400 

500 

600 

700 

800 

900 
1000 
1100 


Dumb-bell  shaped  Water  Spout 
0°Cloud  Level 


160 
140° 
120° 
100° 

80° 

60°' 

Se&.Level 


1200 


FIG.  24.    The  Cottage  City,  Mass.,  waterspout,  August  19,  1896. 


St.  Louis  Tornado,  May  27,  1896. 
FIG.  25.     Illustrating  the  truncated  dumb-bell-shaped  vortex. 


The  vertical  ordinate  is  magnified  ten  times. 


0          100000        200000        300000       400000 

FIG.  26.    The  De  Witte  typhoon,  August  1-3,  1901, 


In  constructing  a  vortex  of  either  type  it  is  necessary  to 
know  two  facts  from  observations:  (1)  the  tangential  velocity 
v  at  a  point  whose  radius  is  S5  in  meters  from  the  axis,  on  a  plane 
defined  by  2  in  the  funnel-shaped  vortex,  or  by  az  in  the  dumb- 


198      CONSTRUCTION  OF  VORTICES   IN  EARTH'S  ATMOSPHERE 

bell-shaped  vortex;  (2)  the  ratio  p,  which  is  the  ratio  of  the 
successive  radii  -or  in  the  tubes.  By  a  series  of  studies  on  the 
spacing  of  the  isobars  in  the  De  Witte  typhoon,  the  St.  Louis 
tornado,  and  the  ocean  cyclone  of  October  11,  1905,  it  was 
found  that  log  p  =  0.20546  seems  to  comply  with  the  positions 
of  the  tubes  in  these  vortices  as  developed  in  the  earth's  atmos- 
phere. In  the  cyclones  and  anticyclones  there  is  a  wide  de- 
parture from  this  simple  constant  ratio,  which  indicates  that 
another  source  of  energy  is  at  work  besides  the  one  generating 
these  simple  vortices,  but  this  will  require  a  fuller  explanation. 
In  Tables  45-53,  and  on  Figs.  24,  25,  26,  are  given  the  results 
of  the  computations  in  sufficient  detail  to  illustrate  the  scope  of 
the  formulas,  the  dimensions  of  the  vortices,  and  the  velocities, 
together  with  the  angles  of  the  helices  which  they  make  in  the 
tubes.  The  funnel-shaped  tube  of  Fig.  24  is  constructed  from 
or  z  in  Table  45,  using  the  tube  (l) ;  the  truncated  dumb-bell- 
shaped  vortex  of  the  St.  Louis  tornado  is  constructed  from  Table 
48.  An  examination  of  the  tables  of  the  velocities  and  the  angles 
suggests  numerous  remarks  on  their  relations,  but  as  they  can 
be  very  clearly  perceived  it  is  not  necessary  to  write  them  down. 
The  vortices  differ  from  one  another  in  their  dimensions,  the 
waterspout,  tornado,  and  hurricane  being  illustrations  of  the 
dumb-bell  vortex.  The  cyclone  shows  a  close  relationship  to 
this  type  of  vortex,  but  it  is  distinctly  modified  by  a  different 
distribution  of  the  thermal  energy.  The  meteorological  data 
are  so  extensive  as  to  make  it  impracticable  to  reproduce  them 
in  this  Treatise. 

Table  45  contains  the  necessary  initial  data  and  the  formulas 
for  developing  a  funnel-shaped  vortex  in  all  its  tubes  from  the 
outer  to  the  inner  in  succession.  Taking  the  assumed  data 
z  =  100,  iff  =  60,  v  =  6.67  m/sec,  \l/  =  vw,  we  proceed  through 
the  formula  to  construct  for  tube  (1),  which  has  the  largest 
radius,  the  constant  of  that  tube  C,  the  velocities  u,  v,  w,  and 
the  horizontal  angle  i  and  the  vertical  angle  y  as  defined  in  Fig. 
14  for  cylindrical  co-ordinates,  and  Fig.  18  for  vortices.  Then 
follows  the  successive  application  of  the  formulas  (535)  to  (550) 
by  which  the  data  of  tubes  (2),  (3),  (4),  (5),  (6)  are  computed. 


EXAMPLES  199 

Thus  by  subtracting  0.20546  from  log  or,  in  succession,  the 
radii  of  the  other  tubes  ttr2,  ttr3j  ztr4,  or5)  ttr6,  are  obtained;  similarly, 
the  log  p  as  indicated  is  to  be  applied  for  C,  u,  v,  w,  i,  y. 

Table  48  contains  the  initial  assumed  data  for  the  dumb-bell- 
shaped  vortex  of  the  St.  Louis  tornado,  truncated  on  the  plane 
az  =  60°.  Thus,  for  az  =  60°,  i  =  -  30°,  v  =  960,  v  =  13.1 
m/sec,  we  find  in  succession,  a  =  0.100,  A,  u,  v,  w,  ttr,  i,  y,  on 
the  tube  (l),  and  on  tubes  (2),  (3),  (4),  (5),  (6),  by  applying 
log  p  or  its  multiples  as  indicated  by  the  working  group  of 
formulas  (535)-(550).  In  the  same  manner  we  proceed  with 
the  De  Witte  typhoon,  the  ocean  cyclone,  and  similar  highly 
developed  vortices.  The  land  cyclone  and  anticyclone  are 
imperfect  vortices,  and  they  involve  a  system  of  hydrodynamic 
stream  lines  which  are  highly  complex  in  their  origin  and  develop- 
ment. It  must  be  constantly  remembered  that  the  important 
radiation  terms  do  not  appear  in  these  vortex  formulas  and 
examples,  so  that  a  fuller  treatment  would  be  much  more  com- 
plex than  the  one  here  briefly  summarized.  A  further  illustra- 
tion will  be  added  in  discussing  the  origin  of  the  cyclone. 

TABLE   45 
THE  COTTAGE  CITY  WATERSPOUT,  AUGUST  19,  1896 

The  Funnel-shaped  Vortex  Tubes,  V  =  C  tffzz 
Collection  of  the  Constants  and  Working  Formulas 
Assumed  data,     z  =  100  meters,  distance  below  cloud  plane 
Iff  —  60  meters,  radius  of  cloud  sheath. 
v  =  6.67  m/sec,  tangential  velocity  at  (z,  tCC). 

$ 

Formulas.     C  =  — o~  the  constant  for  each  tube. 
Wi 

u  —  C  ttr  the  radial  velocity. 

^ 

v  =  —  the  tangential  velocity. 

ttr 

w  =  —  2  C  z  the  vertical  velocity. 
V  =  v  Iff  constant,     log  V  =  2.60206. 

The  ratio  of  the  successive  radii,  P  =         n  '     log  p  =  0.20546. 

n  +  1 

The  successive  radii,         \og'0fn+l  =  logttrn  —  log  p. 
The  successive  velocities,  log  wn  +  i  =  log  un  +  log  P. 
logfn+i  =  logfln  +  log/0, 
log  wn+i  =  log  wn  +  2  log  p. 


200      CONSTRUCTION   OF   VORTICES   IN  EARTH'S   ATMOSPHERE 


The  successive  angles,  log  tan  in  =  constant. 


u 

tan  i  =  — 
v 


log  tan  Jjn+l  =  log  tan  ?„+  log  p.     tan  y  = 

TABLE  46 
THE  VALUES  OF  tcr,  C,  u,  v,  w,  ON  THE  PLANE  z  =  100 


—  w. 

V  sec  i 


Formula 

(1) 

(2) 

(3) 

(4) 

(5) 

(6) 

log  P 

log  or 
•or 

1.77815 
60.0 

1  .  57269 
37.4 

1.36723 
23.3 

1.16177 
14.5 

0.95631 
9.0 

0.75085 
5.6 

l 

0.20546 

log  C 
C 

7.04576 
0.001111 

7.45668 
0.002862 

7.86760 
0.007372 

8.27852 
0.01899 

8.68944 
0.04891 

9.10036 
0.12600 

I   + 

0.41092 

log  M 

U 

8.82391 
0.06667 

9.02937 
0.1070 

9.23483 
0.1717 

9.44029 
0.2756 

9.6*575 
0.4423 

9.85121 
0.7099 

J§    + 

0.20546 

log  i1 

V 

0.82391 
6.67 

1  .  02937 
10.70 

1.23483 
17.17 

1.44029 
27.56 

1.64575 
44.23 

1.85121 
70.99 

1    + 
I 

0.20546 

loga> 
w 

-9.34679 
-0.222 

-9.75771 
-0.572 

-0.16863 
-1.474 

-0.57955 
-3.798 

-0.99047 
-9.783 

-  1.40139 
-25.199 

B 

0.41092 

log  tan  i 
i 

8.00000 
0°     34' 

8.00000 
0°     34' 

8.00000 
0°    34' 

8.00000 
0°     34  ' 

8.00000 
0°     34' 

8.00000 
0°     34' 

s 

o 
U 

log  tan  77 

8.52288 
1     55 

8.72834 
3     4 

8.93380 
4     54 

9  .  13926 
7     51 

9.34472 
12     28 

9.55018 
19     33 

+ 

0.20546 

TABLE  47 
THE  VALUES  OF  ttr,  u,  v,  w,  i,  f\  ON  SEVERAL  PLANES 

The  Radii  Iff 


z 

(1) 

(2) 

(3) 

(4) 

(5) 

(6) 

0 

00 

CO 

CO 

oo 

oo 

00 

1 

600.0 

373.8 

232.9 

145.1 

90.4 

56.3 

•    10 

189.7 

118.2 

73.7 

45.9 

28.6 

17.8 

25 

120.0 

74.8 

46.6 

29.0 

18.1 

11.3 

50 

84.9 

52.9 

32.9 

20.5 

12.8 

8.0 

100 

60.0 

37.4 

23.3 

14.5 

9.0 

5.6 

200 

42.4 

26.4 

16.5 

10.3 

6.4 

4.0 

300 

34.6 

21.6 

13.5 

8.4 

5.2 

3.3 

400 

30.0 

18.7 

11.7 

7.3 

4.5 

2.8 

500 

26.8 

16.7 

10.4 

6.5 

4.0 

2.5 

700 

22.7 

14.1 

8.8 

5.5 

3.4 

2.1 

900 

20.0 

12.5 

7.8 

4.8 

3.0 

1.9 

1100 

18.1 

11.3 

7.0 

4.4 

2.7 

1.7 

EXAMPLES 
TABLE  47. — CONTINUED 


201 


(1) 


(2) 


(3) 


(4) 


(5) 


(6) 


The  Radial  Velocity  u 


0 

1 

00 

0.667 

oo 
1.070 

00 

1.717 

00 

2.756 

00 

4.423 

00 

7.099 

10 

0.211 

0.339 

0.544 

0.874 

1.402 

2.250 

25 

0.133 

0.214 

0.343 

0.551 

0.885 

1.420 

50 

0.094 

0.151 

0.243 

0.390 

0.626 

1.004 

100 

0.067 

0.107 

0.172 

0.276 

0.442 

0.710 

200 

0.047 

0.076 

0.121 

0.195 

0.313 

0.502 

300 

0.039 

0.062 

0.099 

0.159 

0.255 

0.410 

400 

0.033 

0.054 

0.086 

0.138 

0.221 

0.355 

500 

0.030 

0.048 

0.077 

0.123 

0.198 

0.318 

700 

0.025 

0.040 

0.065 

0.104 

0.167 

0.268 

900 

0.022 

0.036 

0.057 

0.092 

0.147 

0.237 

1100 

0.020 

0.032 

0.052 

0.083 

0.133 

0.214 

The  Tangential  Velocity  v 


0 

0 

0 

0 

0 

0 

0 

1 

0.7 

1.1 

1.7 

2.8 

4.4 

7.1 

10 

2.1 

3.4 

5.4 

8.7 

14.0 

22.5 

25 

3.3 

5.4 

8.6 

13.8 

22.1 

35.5 

50 

4.7 

7.6 

12.1 

19.5 

31.3 

50.2 

100 

6.7 

10.7 

17.2 

27.6 

44.2 

71.0 

200 

9.4 

15.1 

24.3 

39.0 

62.6 

100.4 

300 

11.6 

18.5 

29.7 

47.7 

76.6 

122.9 

400 

13.3 

21.4 

34.3 

55.1 

88.5 

142.0 

500 

14.9 

23.9 

38.4 

61.6 

98.9 

158.7 

700 

17.6 

28.3 

45.4 

72.9 

117.0 

187.8 

900 

20.0 

32.1 

51.5 

82.7 

132.7 

213.0 

1100 

22.1 

35.5 

57.0 

91.4 

146.7 

235.7 

The  Vertical  Velocity  w 


0 

0 

0 

0 

0 

0 

0 

1 

-0.0022 

-0.0057 

-  0.0147 

-  0.0380 

-  0.0978 

-  0.2520 

10 

-0.0222 

-0.0572 

-  0.1474 

-  0.3798 

-  0.9783 

-  2.520 

25 

-0.0556 

-0.1431 

-  0.3686 

-  0.9495 

-  2.446 

-  6.300 

50 

-0.1110 

-0.2862 

-  0.7372 

-  1.889 

-  4.891 

-  12.60 

100 

-0.222 

-0.572 

-  1.474 

-  3.798 

-  9.785 

-  25.20 

200 

-0.444 

-1.145 

-  2.949 

-  7.596 

-  19.57 

-  50.40 

300 

-0.667 

-1.717 

-  4.423 

-11.39 

-  29.35 

-  75.60 

400 

-0.889 

-2.290 

-  5.898 

-15.19 

-  39.13 

-100.80 

500 

-1.111 

-2.862 

-  7.372 

-18.99 

-  48.91 

-126.00 

700 

-1.556 

-4.007 

-10.32 

-26.59 

-  68.48 

-176.40 

900 

-2.000 

-5.152 

-13.27 

-34.18 

-  88.05 

-226.80 

1100 

-2.444 

-6.297 

-16.22 

-41.78 

-107.61 

-277.20 

202      CONSTRUCTION  OF  VORTICES  IN  EARTH'S  ATMOSPHERE 


The  Horizontal  Angle  i 


0 

/ 

0 

/ 

o 

90 

0 

Constant  . 

90 

o 

10 

5 

43 

« 

5 

43 

50 

1 

9 

11 

1 

9 

100 

0 

34 

11 

0 

34 

300 

0 

11 

« 

0 

11 

500 

0 

7 

tt 

0 

7 

700 

0 

5 

" 

0 

5 

900 

0 

4 

« 

0 

4 

1100 

0 

3 

0 

3 

The  Vertical  Angle  rj 


0       / 

0       / 

0       / 

0        / 

0         / 

0       / 

0 

0   0 

0   0 

0   0 

0   0 

0   0 

0   0 

10 

0   36 

0   58 

1   33 

2   29 

3   59 

6   22 

50 

1   18 

2   10 

3   28 

5   34 

8   53 

14    5 

100 

1   55 

3    4 

4   54 

7   51 

12   28 

19   33 

300 

3   18 

5   18 

8   28 

13   25 

20   58 

31   35 

500 

4   16 

6   49 

10   52 

17    8 

26   19 

38   26 

700 

5    2 

8    3 

12   48 

20    2 

30   20 

43   12 

900 

5   43 

9    7 

14   27 

22   28 

33   34 

46   48 

1100 

6   19 

10    4 

15   54 

24   34 

36   16 

49   39 

All  the  data  on  the  level  z  =  100  meters  have  been  computed  in  Table  46. 

The  St.  Louis  Tornado,  May  27,  1896 
The  Truncated  Dumb-bell-shaped  Vortex  Tubes,  ip  =  Az  Tff*  sin  az 

TABLE  48 
COLLECTION  OF  THE  CONSTANTS  AND  WORKING  FORMULAS 

Assumed  data  a  z  =  60°     i  =  —  30°  on  the  sea-level  plane. 
ttT  =  960  meters,  radius  of  the  outer  tube. 
v     =13.1  m/sec.  tangential  velocity  at    (ttr .  az). 

a     -  I20ST600  *  °-10°  log  P  =  0.20546. 

log  a  sin  az  =  8.93753        log  a  cos  az  =  8.69897. 

A    = : ,  constant  for  each  tube. 

atCTsm  az 

u     =  —  A  a  tcr  cos  az,  the  radial  velocity. 

aV 
v     =  — ,  the  tangential  velocity. 

w    =  +  A  a  Iff  sin  az,  the  vertical  velocity. 

Ztr  =  ( — : = )  ,  the  radius  on  different  levels. 

\A  a  sm  az) 


EXAMPLES 


203 


tan  *  =  tan  (90  +  az)   =    —  cot  az  =  — . 

v 

tan  ^  = :,  vertical  angle. 

vsect 

q  =  v  sec  i  sec  v,  total  velocity. 

TABLE  49 
THE  VALUES  OF  -or,  A,  u,  v,  w  ON  THE  PLANE  a  2  =  60° 


Formula 

(i) 

(2) 

(3) 

(4) 

(5) 

(6) 

logP 

log  Iff 

2.98227 

2.77681 

2.57135 

2.36589 

2.16043 

1.95497 

-  0.20546 

TV 

960.0 

598.2 

372.7 

232.2 

144.7 

90.2 

log  A 

9.19672 

9.60764 

0.01856 

0.42948 

0.84040 

1.25132 

+  0.41092 

A 

0.1573 

0.4052 

1.0437 

2  .  6883 

6.9247 

17.8371 

f 

log  « 

-0.87796 

-1.08342 

-1.28888 

-1.49434 

-1.69980 

-1.90526 

Y   -  0.20546 

u 

-7.6 

-12.1 

-19.5 

-31.2 

-50.1 

-80.4 

*G 

8 

log  v 

1.11652 

1.32198 

1.52744 

1.73290 

1.93836 

2  .  14382 

~  +  0.20546 

V 

13.1 

21.0 

33.7 

54.1 

86.8 

139.3 

1 

log  w 

9.43528 

9.84620 

0.25712 

0.66804 

1  .  07896 

1.48988 

§    -f  0.41092 

w 

0.27 

0.70 

1.81 

4.66 

12.00 

30.89 

«M 

8 

log  tan  t 

—9.76144 

Constant 

—9.76144 

a 

—30° 

—30° 

u  ' 

log  tan  ?/ 

8.25629 

8.46175 

8.66721 

8.87267 

9.07813 

9.28359 

+  0.20546 

? 

1°     2' 

1°     39' 

2°     40' 

4°     16' 

6°     50' 

10°     53' 

THE 


TABLE  50 
VALUES  OF  iff,  u,  v,  w,  i,  ~n  ON  SEVERAL  PLANES 


a  z 

(1) 

(2) 

(3) 

(4) 

(5) 

(6) 

The  Radii 


180° 

00 

00 

00 

00 

00 

00 

170 

2143.9 

1335.8 

832.3 

518.6 

323.1 

201.3 

160 

1527.6 

951.8 

593.0 

369.5 

230.2 

143.5 

150 

1263.4 

787.2 

490.5 

305.6 

190.4 

118.6 

140 

1114.3 

694.3 

432.6 

269.5 

167.9 

104.6 

130 

1020.7 

636.0 

396.3 

246.9 

153.8 

95.9 

120 

960.0 

598.1 

372.7 

232.2 

144.7 

90.2 

110 

921.6 

574.2 

357.8 

222.9 

138.9 

86.5 

100 

900.2 

560.9 

349.5 

217.8 

135.7 

84.5 

90 

893.4 

556.6 

346.8 

216.1 

134.6 

83.9 

80 

900.2 

560.9 

349.5 

217.8 

135.7 

84.5 

70 

921.6 

574.2 

357.8 

222.9 

138.9 

86.5 

60 

960.0 

598.1 

372.7 

232.2 

144.7 

90.2 

204      CONSTRUCTION   OF   VORTICES   IN  EARTH'S   ATMOSPHERE 
TABLE  50.— CONTINUED 


(1) 


(2) 


(3) 


(4) 


(5) 


(6) 


The  Radial  Velocity  u 


180° 

00 

00 

00 

00 

00 

00 

170 

33.2 

53.3 

85.5 

137.3 

220.3 

353.6 

160 

22.6 

36.2 

58.2 

93.4 

149.8 

240.4 

150 

17.2 

27.6 

44.3 

71.2 

114.2 

183.3 

140 

13.4 

21.6 

34.6 

55.5 

89.1 

143.0 

130 

10.3 

16.6 

26.6 

42.7 

68.5 

109.9 

120 

7.6 

12.1 

19.5 

31.2 

50.1 

80.4 

110 

5.0 

8.0 

12.8 

20.5 

32.9 

52.8 

100 

2.5 

4.0 

6.3 

10.2 

16.3 

26.2 

90 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

80 

-2.5 

-  4.0 

-  6.3 

-10.2 

-16.3 

-26.2 

70 

-5.0 

-  8.0 

-12.8 

-20.5 

-32.9 

-52.8 

60 

-7.6 

-12.1 

-19.5 

-31.2 

-50.1 

-80.4 

The  Tangential  Velocity  v 


180° 

0 

0 

0 

0 

0 

0 

170 

5.9 

9.4 

15.1 

24.2 

38.9 

62.4 

160 

8.2 

13.2 

21.2 

34.0 

54.5 

87.5 

150 

9.9 

16.0 

25.6 

41.1 

65.9 

105.8 

140 

11.3 

18.1 

29.0 

46.6 

74.8 

120.0 

130 

12.3 

19.7 

31.7 

50.9 

81.6 

131.0 

120 

13.1 

21.0 

33.7 

54.1 

86.8 

139.3 

110 

13.6 

21.9 

35.1 

56.3 

90.4 

145.1 

100 

13.9 

22.4 

35.9 

57.7 

92.5 

148.5 

90 

14.1 

22.6 

36.2 

58.1 

93.2 

149.6 

80 

13.9 

22.4 

35.9 

57.7 

92.5 

148.5 

70 

13.6 

21.9 

35.1 

56.3 

90.4 

145.1 

60 

13.1 

21.0 

33.7 

54.1 

86.8 

139.3 

The  Vertical  Velocity  -w 


180° 

0 

0 

0 

0 

0 

0 

170 

0.06 

0.14 

0.36 

0.93 

2.41 

6.20 

160 

0.11 

0.28 

0.71 

1.84 

4.74 

12.20 

150 

0.16 

0.41 

1.04 

2.69 

6.93 

17.84 

140 

0.20 

0.52 

1.34 

3.46 

8.90 

22.93 

130 

0.24 

0.62 

1.60 

4.22 

10.86 

27.96 

120 

0.27 

0.70 

1.81 

4.66 

11.99 

30.90 

110 

0.30 

0.76 

1.96 

5.05 

13.01 

33.52 

100 

0.31 

0.80 

2.06 

5.30 

13.64 

35.13 

90 

0.32 

0.81 

2.09 

5.38 

13.85 

35.67 

80 

0.31 

0.80 

2.06 

5.30 

13.64 

35.13 

70 

0.30 

0.76 

1.96 

5.05 

13.01 

33.52 

60 

0.27 

0.70 

1.81 

4.66 

11.99 

30.90 

EXAMPLES 
The  Horizontal  Angle  i 


205 


180° 

+90° 

Constant  

+90° 

160 

+70 

+70 

140 

+50 

« 

+50 

120 

+30  . 

« 

+30 

100 

+  10 

« 

+10 

90 

0 

a 

0 

80 

-10 

H 

-10 

70 

-20 

II 

-20 

60 

-30 

II 

-30 

The  Vertical  Angle  rj 

180° 

0°    0' 

0°    0' 

0°    0' 

0°    0' 

0°    0' 

0°  0' 

170 

0    6 

0    9 

0    14 

0    23 

0    37 

0  59 

160 

0    15 

0    25 

0    40 

1    4 

1    42 

2  44 

150 

0    27 

0    44 

1    10 

1    52 

3    0 

4  49 

140 

0    40 

1    4 

1    42 

2    44 

4    23 

7   0 

130 

0    52 

1    23 

2    13 

3    38 

5    49 

9  17 

120 

2 

1    39 

2    40 

4    16 

6    50 

10  53 

110 

10 

1    52 

3    0 

4    49 

7    42 

12  15 

100 

15 

2     1 

3    14 

5    10 

8    16 

13   7 

90 

17 

2    3 

3    18 

5    17 

8   27 

13  25 

80 

15 

2     1 

3    14 

5    10 

8    16 

13   7 

70 

10 

1    52 

3    0 

4    49 

7    42 

12  15 

60 

1    2 

1    39 

2    40 

4    16 

6    50 

10  53 

The  data  on  the  level  a  z  have  been  computed  in  Table  49. 


TABLE   51 

THE  DE  WITTE  TYPHOON,  AUGUST  1-3,  1901, 
IN  THE  CHINA  SEA 

Results  from  the  plane  a  z  =  60° 


Initial  z  =  12000  meters  a  = 


180C 


12000  +  600 


=  0.010 


z 

(1) 

(2) 

(3) 

(4) 

(5) 

(6) 

A 

.002016 

.005193 

.013375 

.034452 

.088744 

.248922 

206      CONSTRUCTION  OF  VORTICES   IN  EARTH'S  ATMOSPHERE 
The  Radii  W  in  Meters 


a  z  =  180° 

00 

00 

00 

00 

00 

00 

170 

887600 

534338 

332938 

207443 

129253 

80534 

160 

611071 

380742 

237232 

147813 

92098 

57384 

150 

505389 

314900 

196205 

122250 

76172 

47460 

140 

445740 

277727 

173044 

107820 

67180 

41858 

130 

408310 

254406 

158515 

98766 

61539 

38344 

120 

384018 

239272 

149083 

92890 

57878 

36062 

110 

368655 

229700 

143120 

89174 

55563 

34619 

100 

360117 

224379 

139803 

87108 

54275 

33818 

90 

357367 

222663 

138735 

86444 

53860 

33559 

80 

360117 

224379 

139803 

87108 

54275 

33818 

70 

368655 

229700 

143120 

89174 

55563 

34619 

60 

384018 

239272 

149083 

92890 

57878 

36062 

The  Radial  Velocity  u  in  Meters  per  Second 


az  =  180° 

00 

00 

00 

00 

00 

00 

170 

17.03 

27.32 

43.86 

70.38 

112.96 

181.30 

160 

11.58 

18.58 

29.82 

47.85 

76.80 

123.27 

150 

8.82 

14.16 

22.72 

36.47 

58.53 

93.94 

140 

6.88 

11.05 

17.73 

28.46 

45.67 

73.30 

130 

5.29 

8.49 

13.63 

21.87 

35.10 

56.34 

120 

3.87 

6.21 

9.97 

16.00 

25.68 

41.22 

110 

2.54 

4.08 

6.55 

10.51 

16.86 

27.07 

100 

1.28 

2.02 

3.25 

5.21 

8.36 

13.42 

90 

0.00 

0.00 

0.00 

0.00 

0.00 

0.00 

80 

-1.28 

-2.02 

-3.25 

-  5.21 

-  8.36 

-13.42 

70 

-2.54 

-4.08 

-6.55 

-10.51 

-16.86 

-27.07 

60 

-3.87 

-6.21 

-9.97 

-16.00 

-25.68 

-41.22 

The  Tangential  Velocity  v  in  Meters  per  Second 


a  z  =  180° 

0.00 

0.00 

0.00 

0.00 

0.00 

0.00 

170 

3.00 

4.82 

7.73 

12.41 

19.92 

31.97 

160 

4.21 

6.76 

10.85 

17.42 

27.95 

44.86 

150 

5.09 

8.17 

13.12 

21.06 

33.79 

54.24 

140 

5.78 

9.27 

14.88 

23.88 

38.32 

61.50 

130 

6.31 

10.12 

16.24 

26.07 

41.84 

67.14 

120 

6.70 

10.76 

17.27 

27.72 

44.48 

71.39 

110 

6.98 

11.21 

17.99 

28.87 

46.34 

74.36 

100 

7.15 

11.47 

18.42 

29.55 

47.43 

76.13 

90 

7.20 

11.56 

18.56 

29.78 

47.80 

76.71 

80 

7.15 

11.47 

18.42 

29.55 

47.43 

76.13 

70 

6.98 

11.21 

17.99 

28.87 

46.34 

74.36 

60 

6.70 

10.76 

17.27 

27.72 

44.48 

71.39 

THE  OCEAN  AND  THE  LAND  CYCLONES 


The  Vertical  Velocity  iv  in  Meters  per  Second 


207 


a  z  =  180° 

0.0000 

0.0000 

0.0000 

0.0000 

0.0000 

0.0000 

170 

0.0007 

0.0018 

0.0046 

0.0120 

0.0308 

0.0794 

160 

0.0014 

0.0036 

0.0091 

0.0236 

0.0607 

0.1564 

150 

0.0020 

0.0052 

0.0134 

0.0345 

0.0887 

0.2286 

140 

0.0026 

0.0067 

0.0172 

0.0443 

0.1141 

0.2939 

130 

0.0031 

0.0080 

0.0205 

0.0528 

0.1340 

0.3502 

120 

0.0035 

0.0090 

0.0232 

0.0597 

0.1537 

0.3959 

110 

0.0038 

0.0098 

0.0251 

0.0648 

0.1668 

0.4296 

100 

0.0040 

0.0102 

0.0263 

0.0679 

0.1748 

0.4502 

90 

0.0040 

0.0104 

0.0268 

0.0689 

0.1775 

0.4572 

80 

0.0040 

0.0102 

0.0263 

0.0679 

0.1748 

0.4502 

70 

0.0038 

0.0098 

0.0251 

0.0648 

0.1668 

0.4296 

60 

0.0035 

0.0090 

0.0232 

0.0597 

0.1537 

0.3959 

The  Ocean  and  the  Land  Cyclones 

The  tornadoes  and  hurricanes  always  occur  in  strata  of  air 
which  are  practically  quiescent  in  the  vertical  direction,  the 
tornadoes  in  the  lower  levels  of  stagnant  air  during  hot  summer 
afternoons,  and  the  hurricanes  in  the  neighborhood  of  the 
latitudes  30°  to  35°,  where  the  east  and  west  movements  in  the 
general  circulation  practically  disappear.  Should  hurricanes 
move  into  higher  latitudes,  where  the  eastward  drift  prevails 
with  an  increase  of  its  velocity  in  proportion  to  the  height  above 
the  ocean,  the  nearly  perfect  dumb-bell  vortices  which  represent 
them  are  transformed  into  imperfect  vortices  of  the  same  general 
type.  The  penetration  of  the  head  of  the  vortex  into  the  midst 
of  the  eastward  drift  introduces  components  of  resistance  which 
deplete  and  even  destroy  the  type  in  the  upper  levels,  so  that 
it  is  degraded  to  a  cyclone,  or  imperfect  dumb-bell  vortex  by  the 
mere  mechanical  action.  Furthermore,  the  temperature  dis- 
tribution is  distinctly  different  in  hurricanes  and  in  cyclones. 
In  the  former  the  temperature  differences  are  separated  by 
horizontal  planes  in  the  upper  levels,  while  in  the  latter  the 
temperatures  are  separated  chiefly  in  a  vertical  direction.  The 
hurricanes  have  a  symmetrical  horizontal  distribution  of  tem- 
peratures, but  in  cyclones  the  temperature  distribution  is  de- 
cidedly asymmetrical,  as  is  well  known  from  the  weather  maps 
on  the  surface.  The  same  asymmetry  of  temperature  continues 


208      CONSTRUCTION  OF  VORTICES   IN  EARTH'S  ATMOSPHERE 

to  the  levels  as  high  as  10,000  meters,  warm  on  the  east  and  cold 
on  the  west  side  of  cyclones  in  the  United  States.  These  broad, 
thin  sheets  of  warm  and  cold  air,  under  the  action  of  gravity, 
tend  to  return  to  a  horizontal  symmetry  by  the  production  of 
stream  lines,  whereby  the  cold  air  underruns  the  warm  stream 
to  the  east  and  to  the  west  by  dividing  into  two  branches,  while 
the  warm  air  overruns  the  cold  air  to  the  east  and  to  the  west 
similarly  in  two  branches.  Complicated  stream  lines  are  thus 
produced,  which  are  those  observed  in  the  free  air,  after  entering 
into  composition  with  the  velocities  of  the  general  circulation 
of  the  locality.  This  complex  subject  will  require  much  more 
study  than  has  been  possible  up  to  the  present  time  in  order  to 
secure  a  complete  analysis  of  the  data,  but  it  is  clear  that  the 
research  must  proceed  along  certain  lines  which  can  be  briefly 
indicated. 

The  first  problem  is  to  separate  the  imperfect  from  the 
perfect  vortices,  and  to  assign  the  components  of  resistance, 
that  is,  to  construct  a  reverse  vortex  which  is  practically  equiva- 
lent to  the  system  of  reactions  that  prevents  the  dumb-bell  vortex 
from  developing  into  a  pure  form.  The  second  problem  is  to 
determine  the  stream  lines  by  which  the  masses  of  air  at  different 
temperatures  are  drawn  by  the  force  of  gravity  into  these  im- 
perfect cyclonic  vortices.  The  ocean  cyclone,  October  11,  1905, 
has  been  taken  to  illustrate  the  composition  of  vortices,  and 
the  land  cyclones  must  be  studied  more  at  length  from  the  data 
provided  by  balloon  and  kite  ascensions  in  Europe  and  the 
United  States.  The  ocean  cyclone  is  more  highly  developed 
than  the  land  cyclone,  and  affords  a  convenient  transition  from 
the  hurricane  to  the  ordinary  cyclonic  storm.  The  cyclone  of 
October  11,  1905,  has  been  reduced  to  an  equivalent  cylindrical 
vortex  by  taking  the  mean  radii  as  measured  in  four  directions 
at  right  angles  to  each  other.  This  mechanical  process  need 
not  be  repeated  here,  but  the  result  is  that  the  radii  are  not 
spaced  in  the  vortical  geometrical  ratio.  They  diverge  from  that 
model  which  belongs  to  the  perfect  vortex.  The  corresponding 
velocities  tangential  to  the  equivalent  circular  isobars  were 
constructed  from  the  observed  values  in  different  parts  of  the 


THE    OCEAN  AND   THE   LAND   CYCLONES 


209 


cyclone  as  reported  by  the  110  vessels  that  made  observations 
on  that  date. 

TABLE  52 

THE  OCEAN  CYCLONE,  OCTOBER  11,  1905 

The  Imperfect  Dumb-bell-shaped  Vortex  V'l 

Results  for  the  Plane  a  z  =  60° 

180° 
Initial  z  =  8000  meters,  a  =  §QQQ  +  4QQQ  =  0.015,  log  p  =0.10600. 


z 

(1) 

(2) 

(3) 

(4) 

(5) 

(6) 

(7) 

(8) 

A, 

.00089 

.00127 

.00184 

.00265 

.00382 

.00551 

.00795 

.01146 

The  Radii  Iff  in  Kilometers 


0 

a  z  =  180 

00 

00 

00 

00 

00 

00 

00 

00 

170 

3079.9 

2412.9 

1890.3 

1480.6 

1160.2 

909.0 

712.1 

557.9 

160 

2194.6 

1719.3 

1346.9 

1055.2 

826.7 

647.7 

507.4 

397.5 

150 

1815.0 

1422.0 

1114.0 

872.8 

683.7 

535.7 

419.6 

328.8 

.  140 

1600.8 

1254.1 

982.5 

769.7 

603.0 

472.4 

370.1 

290.0 

130 

1466.4 

1148.8 

900.0 

705.1 

552.4 

432.8 

339.0 

265.6 

120 

1379.2 

1080.5 

846.5 

663.2 

519.5 

407.0 

318.9 

249.8 

110 

1324.0 

1037.2 

812.6 

636.6 

498.8 

390.7 

306.1 

239.8 

100 

1293.3 

1013.2 

793.8 

621.9 

487.2 

381.7 

299.0 

234.3 

90 

1283.4 

1005  .  5 

787.7 

617.1 

483.5 

378.8 

296.7 

232.5 

80 

1293.3 

1013.2 

793.8 

621.9 

487.2 

381.7 

299.0 

234.3 

70 

1324.0 

1037.2 

812.6 

636.6 

498.8 

390.7 

306.1 

239.8 

60 

1379.2 

1080.5 

846.5 

663.2 

519.5 

407.0 

318.9 

249.8 

The  Radial  Velocity  u\  in  Meters  per  Second 


a  z  =  180 

00 

00 

00 

00 

00 

00 

00 

00 

170 

40.2 

45.4 

51.3 

58.0 

65.5 

74.0 

83.6 

94.4 

160 

27.3 

30.9 

34.9 

39.4 

44.5 

50.3 

56.8 

64.2 

150 

20.8 

23.5 

26.6 

30.0 

33.9 

38.3 

43.0 

48.9 

140 

16.2 

18.4 

20.7 

23.4 

26.5 

29.9 

33.8 

38.1 

130 

12.5 

14.1 

16.0 

18.0 

20.4 

23.0 

26.0 

29.4 

120 

9.1 

10.3 

11.7 

13.2 

14.9 

16.8 

19.0 

21.5 

110 

6.0 

6.8 

7.7 

8.7 

9.8 

11.0 

12.5 

14.1 

100 

3.0 

3.4 

3.8 

4.3 

4.8 

5.5 

6.2 

7.0 

90 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

80 

-  3.0 

-  3.4 

-  3.8 

-  4.3 

-  4.8 

-  5.5 

-  6.2 

-  7.0 

70 

-  6.0 

-  6.8 

-  7.7 

-  8.7 

-  9.8 

-11.0 

-12.5 

-14.1 

60 

-  9.1 

-10.3 

-11.7 

-13.2 

-14.9 

-16.8 

-19.0 

-21.5 

210      CONSTRUCTION  OF  VORTICES  IN  EARTH'S  ATMOSPHERE 


The  Tangential  Velocity  vi  in  Meters  per  Second 


az  =  180 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

170 

7.1 

8.0 

9.0 

10.2 

11.6 

13.0 

14.7 

16.7 

160 

9.9 

11.2 

12.7 

14.3 

16.2 

18.3 

20.7 

23.4 

150 

12.0 

13.6 

15.4 

17.3 

19.6 

22.1 

25.0 

28.3 

140 

13.6 

15.4 

17.4 

19.7 

22.2 

25.1 

28.4 

32.0 

130 

14.9 

16.8 

19.0 

21.5 

24.3 

27.4 

31.0 

35.0 

120 

15.8 

17.9 

20.2 

22.8 

25.8 

29.1 

32.9 

37.2 

110 

16.5 

18.6 

21.0 

23.8 

26.9 

30.3 

34.3 

38.7 

100 

16.9 

19.1 

21.5 

24.3 

27.5 

31.1 

35.1 

39.7 

90 

17.0 

19.2 

21.7 

24.5 

27.7 

31.3 

35.4 

40.0 

80 

16.9 

19.1 

21.5 

24.3 

27.5 

31.1 

35.1 

39.7 

70 

16.5 

18.6 

21.0 

23.8 

26.9 

30.3 

34.3 

38.7 

60 

15.8 

17.9 

20.2 

22.8 

25.8 

29.1 

32.9 

37.2 

The  Vertical  Velocity  w\  in  Meters  per  Second 


o 

az  =  180 

.0000 

.0000 

.0000 

.0000 

.0000 

.0000 

.0000 

.0000 

170 

.0003 

.0004 

.0006 

.0009 

.0013 

.0019 

.0028 

.0040 

160 

.0006 

.0009 

.0013 

.0018 

.0026 

.0038 

.0054 

.0078 

150 

.0009 

.0013 

.0018 

.0026 

.0038 

.0055 

.0079 

.0115 

140 

.0011 

.0016 

.0024 

.0034 

.0049 

.0071 

.0102 

.0147 

130 

.0014 

.0020 

.0028 

.0041 

.0059 

.0084 

.0122 

.0176 

120 

.0015 

.0022 

.0032 

.0046 

.0066 

.0095 

.0138 

.0198 

110 

.0017 

.0024 

.0035 

.0050 

.0072 

.0104 

.0149 

.0215 

100 

.0017 

.0025 

.0036 

.0052 

.0075 

.0109 

.0157 

.0226 

90 

.0018 

.0025 

.0037 

.0053 

.0076 

.0110 

.0159 

.0229 

80 

.0017 

.0025 

.0036 

.0052 

.0075 

.0109 

.0157 

.0226 

70 

.0017 

.0024 

.0035 

.0050 

.0072 

.0104 

.0149 

.0215 

60 

.0015 

.0022 

.0032 

.0046 

.0066 

.0095 

.0138 

.0198 

TABLE  53 

THE  OCEAN  CYCLONE,  OCTOBER  11,  1905 
The  Perfect  Dumb-bell-Shaped  Vortex  Vo 
The  Radii  ZCT  remain  the  Same 


z 

A, 

(1) 
.00089 

(2) 
.00144 

(3) 
.00235 

(4) 
.00382 

(5) 
.00623 

(6) 
.01014 

(7) 
.01653 

(8) 
.02699 

The  Radial  Velocity  «0 


0 

a  z  =  180 

00 

00 

00 

00 

oo 

00 

00 

00 

170 

40.2 

51.3 

65.5 

83.6 

106.7 

136.2 

173.8 

221.9 

160 

27.3 

34.9 

44.5 

56.8 

72.5 

92.6 

118.2 

150.9 

150 

20.8 

26.6 

33.9 

43.3 

55.3 

70.6 

90.1 

115.0 

140 

16.2 

20.7 

26.5 

33.8 

43.1 

55.1 

70.3 

89.7 

130 

12.5 

15.9 

20.4 

26.0 

33.2 

42.3 

54.0 

69.0 

120 

9.1 

11.7 

14.9 

19.0 

24.3 

31.0 

39.5 

50.4 

110 

6.0 

7.7 

9.7 

12.5 

15.9 

20.3 

26.0 

33.1 

100 

3.0 

3.8 

4.8 

6.2 

7.9 

10.1 

12.9 

16.4 

90 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

THE  OCEAN  AND  THE  LAND  CYCLONES 


211 


The  Tangential  Velocity  v0 


az  =  180 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

170 

7.1 

9.0 

11.6 

14.7 

18.8 

24.0 

30.7 

39.1 

160 

9.9 

12.7 

16.2 

20.7 

26.4 

33.7 

43.0 

54.9 

150 

12.0 

15.4 

19.6 

25.0 

31.9 

40.8 

52.0 

66.4 

140 

13.6 

17.4 

22.2 

28.4 

36.2 

46.2 

59.0 

75.3 

130 

14.9 

19.0 

24.2 

31.0 

39.5 

50.4 

64.4 

82.2 

120 

15.8 

20.2 

25.8 

32.8 

42.0 

53.6 

68.5 

87.4 

110 

16.5 

21.0 

26.9 

34.3 

43.8 

55.9 

71.3 

91.0 

100 

16.9 

21.5 

27.5 

35.1 

44.8 

57.2 

73.0 

93.2 

90 

17.0 

21.7 

27.7 

35.4 

45.1 

57.6 

73.6 

93.9 

The  Vertical  Velocity 


a  z  =  180 

.0000 

.0000 

.0000 

.0000 

.0000 

.0000 

.0000 

.0000 

170 

.0003 

.0005 

.0008 

.0013 

.0022 

.0035 

.0057 

.0094 

160 

.0006 

.0010 

.0016 

.0026 

.0043 

.0069 

.0113 

.0184 

150 

.0009 

.0014 

.0024 

.0038 

.0062 

.0101 

.0165 

.0269 

140 

.0011 

.0018 

.0030 

.0049 

.0080 

.0130 

.0212 

.0346 

130 

.0014 

.0022 

.0036 

.0058 

.0095 

.0155 

.0253 

.0412 

120 

.0015 

.0025 

.0041 

.0066 

.0108 

.0176 

.0286 

.0466 

110 

.0017 

.0027 

.0044 

.0072 

.0117 

.0191 

.0311 

.0506 

100 

.0017 

.0028 

.0046 

.0075 

.0123 

.0202 

.0326 

.0530 

90 

.0018 

.0029 

.0047 

.0076 

.0124 

.0203 

.0331 

.0538 

TABLE  54 

THE  REVERSING  OR  COMPONENT  VORTEX,  Vi— 


At.  =  Ai  —  Ao 

.00000     -.00017 

-.00051 

-.00117 

-.00241 

-.00463 

-.00858 

-.01553 

The  radial  velocity  of  the  reverse  vortex,   uz 


az  =  180° 

0.0 

170 

0.0 

-5.9 

-14.2 

-25.6 

-41.2 

-62.2 

-90.2 

-127.5 

160 

0.0 

-4.0 

-  9.6 

-17.4 

-28.0 

-42.3 

-61.4 

-  86.7 

150 

0.0 

-3.1 

-  7.3 

-13.3 

-21.4 

-32.3 

-47.1 

-  66.1 

140 

0.0 

-2.3 

-  5.8 

-10.4 

-16.6 

-25.2 

-36.5 

-  51.6 

130 

0.0 

-1.8 

-  4.4 

-  8.0 

-12.8 

-19.3 

-28.0 

-  39.6 

120 

0.0 

-1.4 

-  3.2 

-  5.8 

-  9.4 

-14.2 

-20.5 

-  28.9 

110 

0.0 

-0.9 

-  2.0 

-  3.8 

-  6.1 

-  9.3 

-13.5 

-  19.0 

100 

0.0 

-0.4 

-  1.0 

-  1.9 

-  3.1 

-  4.6 

-  6.7 

-  9.4 

90 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

The  tangential  velocity  of  the  reverse  vortex, 


a  z  =  180° 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

0.0 

170 

0.0 

-1.0 

-2.6 

-  4.5 

-  7.2 

-11.0 

-16.0 

-22.4 

160 

0.0 

-1.5 

-3.5 

-  6.4 

-10.2 

-15.4 

-22.3 

-31.5 

150 

0.0 

-1.8 

-4.2 

-  7.7 

-12.3 

-18.7 

-27.0 

-38.1 

140 

0.0 

-2.0 

-4.8 

-  8.7 

-14.0 

-21.1 

-30.6 

-43.3 

130 

0.0 

-2.2 

-5.2 

-  9.5 

-15.2 

-23.0 

-33.4 

-47.2 

120 

0.0 

-2.3 

-5.6 

-10.0 

-16.2 

-24.5 

-35.6 

-50.2 

110 

0.0 

-2.4 

-5.9 

-10.5 

-16.9 

-25.6 

-37.0 

-52.3 

100 

0.0 

-2.4 

-6.0 

-10.8 

-17.3 

-26.1 

-37.9 

-53.5 

90 

0.0 

-2.5 

-6.0 

-10.9 

-17.4 

-26.3 

-38.2 

-53.9 

212      CONSTRUCTION   OF   VORTICES   IN  EARTH'S   ATMOSPHERE 


The  vertical  velocity  of  the  reverse  vortex, 


az  =  180° 

.0000 

.0000 

.0000 

.0000 

.0000 

.0000 

,0000 

.0000 

170 

.0000 

-.0001 

-  .  0002 

-  .  0004 

-  .  0009 

-.0016 

-  .  0029 

-  .  0054 

160 

.0000 

-.0001 

-  .  0003 

-  .  0008 

-.0017 

-.0031 

-.0059 

-.0106 

150 

.0000 

-.0001 

-  .  0005 

-.0012 

-  .  0024 

-.0046 

-  .  0086 

-.0154 

140 

.0000 

-.0002 

-  .  0006 

-.0015 

-  .  0031 

-.0059 

-.0110 

-.0199 

130 

.0000 

-  .  0002 

-.0008 

-.0017 

-  .  0036 

-  .  O0'<  1 

-.0131 

-  .  0236 

120 

.0000 

-  .  0003 

-  .  0009 

-  .  0020 

-  .  0042 

-.0081 

-.0148 

-  .  0268 

110 

.0000 

-  .  0003 

-  .  0009 

-  .  0022 

-.0045 

-  .  0087 

-.0162 

-.0291 

100 

.0000 

-.0003 

-.0010 

-  .  0023 

-  .  0047 

-  .  0091 

-.0169 

-  .  0301 

90 

.0000 

-  .  0004 

-.0010 

-.0023 

-  .  0048 

-.0093 

-  .  0172 

-.0309 

The  Composition  of  Vortices 

By  subtracting  the  computed  velocities  of  the  perfect  vortex 
from  those  of  the  imperfect  vortex,  u\  —  UQ  =  u2,  Vi  —  VQ  =  v2, 
Wi  —  WQ  =  w2,  we  have  a  component  vortex  which,  added  to 
the  perfect  vortex,  will  produce  the  observed  imperfect  vortex, 
u\  —  UQ  -f  ^2,  v\  =  VQ  +  v2,  Wi  =  w0  -\-  w2,  the  signs  being  added 
algebraically.  The  corresponding  values  of  the  constants  A, 
AI  —  AQ  =  A2,  can  be  found  by  computing  the  values  of  A  for 
the  derived  velocities  by  the  formulas, 


(643)     A2  = 


U2 


-W2 


a  w  cos  a  z 


sin  a  z       2  sin  a  z' 


More  simply,  the  algebraic  values  of  A2  are  derived  im- 
mediately from  AI  (imperfect  vortex)  —  A0  (perfect  vortex), 
whence  the  corresponding  velocities  u2,  v2,  w2  can  be  computed 
in  the  usual  manner. 

Table  52  gives  the  values  of  AI,  MI,  Vi,  w\  in  the  imperfect 
vortex;  Table  53  those  of  A0,  u0,  VQ,  wd  in  the  perfect  vortex, 
and  Table  54  those  of  A2j  «2,  v2,  w2  in  the  component  reversing 
vortex.  A  comparison  of  the  velocities  in  these  tables  shows 
that,  by  starting  with  the  same  radius  and  velocity  on  the  outer 
isobar  (1),  the  observed  imperfect  vortex  departs  more  and  more 
from  the  corresponding  perfect  vortex  in  proportion  as  the  veloci- 
ties approach  the  axis.  The  component  vortex  which  is  equiva- 
lent to  these  differences  is  a  vortex  reversed  in  all  respects  to  the 
original  vortex,  revolving  in  the  opposite  direction  and  directed 
downward  from  the  clouds  to  the  surface  of  the  sea.  This 
principle  of  the  composition  of  vortices  through  the  constants  A 


THE    REVERSED   DUMB-BELL  VORTEX 


213 


of  the  successive  tubes  is  very  important,  and  leads  to  many 
practical  researches  in  the  theory  of  cyclones,  because  it  enables 
us  to  take  account  of  the  numerous  departures  from  the  pure 
vortex  law,  without  giving  up  the  advantages  of  the  method  of 
vortex  computations. 

The  Reversed  Dumb-bell  Vortex 

A  very  erroneous  impression  would  be  left  if  it  were  sup- 
posed that  the  imperfect  dumb-bell  vortex  could  be  applied 
directly  to  the  study  of  the  common  cyclones  in  the  atmosphere. 
The  dumb-bell  vortex  seems  to  be  essentially  reversed,  turned 


Height 

in 
meters 


(6)  (5)  (4)  (3)  (2)  (I) 


10000 
0000 
8000 
7000 
6000 
5000 
1000 
3000 
2000 
1000 
000 


11500 


FIG.  27.  The  constant  a  A  lines  as  derived  from  observations  in  the  land  cyclones. 

inside  out,  as  can  be  seen  by  Table  55,  and  Fig.  27.  The  dis- 
cussion of  the  velocity  components  resulted  as  given  in  the 
Cloud  Report,  in  Table  126.  Taking  the  velocities  in  that 
table  and  plotting  them  on  diagrams,  a  consistent  system  was 
deduced  which  conforms  on  the  average  to  that  there  given.  It 


214      CONSTRUCTION   OF  VORTICES  IN  EARTH'S  ATMOSPHERE 


is  produced  on  Table  55,  I,  II,  for  every  1,000-meter  level,  and 
for  the  radial  distances  given  in  the  normal  land  cyclone.  Since 

tan  a  z  =  — ,  by  (527)  and  (528),  we  can  compute  a  A  for  each 

level  and  tube.  The  result  appears  in  Table  55,  III.  On  Fig. 
27,  at  the  radial  distances,  80,  300,  508,  733,  975, 1,250  kilometers, 
these  values  of  a  A  were  plotted  down,  and  lines  of  equal  a  A 
were  drawn  and  they  are  given  on  Fig.  27.  They  contain  the 
surprising  result  that  the  radial  distances  W  are  arranged  nearly 
on  a  geometrical  ratio  system,  as  can  be  readily  seen*  by  making 
the  tests,  and  that  the  old  ratio  value  log  p  =  0.20546  is  quite 
competent  to  satisfy  the  average  conditions.  The  lines  a  A 
are,  however,  concave  towards  the  axis,  instead  of  convex  as  in 
the  hurricane,  and  the  lines  are  closed  up  on  the  outer  circles 
rather  than  on  the  inner,  this  being  a  complete  reversal  of  the 
configuration.  The  theoretical  and  the  thermodynamical  con- 
ditions which  produce  this  circulating  structure  have  been 
indicated  in  the  series  of  papers  on  the  "  Thermodynamics  of  the 
Atmosphere,"  W.  B.  372,  1907.  The  subject  will  require  further 
study  and  investigation. 

TABLE  55 

THE  OBSERVED  RADIAL  AND  TANGENTIAL  VELOCITIES  IN  CYCLONES. 
INTERNATIONAL  CLOUD  REPORT 

I.  u  =  the  observed  radial  velocities 


Height 
in  Meters 

(i) 
1250000 

(2) 

975000 

(3) 

733000 

(4) 

508000 

(5) 
300000 

(6) 

80000 

10000 

-0.5 

-0.7 

-1.0 

-1.3 

-1.6 

-2.0 

9000 

-1.5 

-2.1 

-2.7 

-3.0 

-3.7 

-4.5 

8000 

-2.0 

-3.0 

-4.0 

-4.5 

-5.5 

-6.0 

7000 

-3.0 

-4.0 

-5.0 

-6.0 

-7.0 

-7.5 

6000 

-3.0 

-4.0 

-5.0 

-6.0 

-7.0 

-8.0 

5000 

-2.5 

-3.0 

-4.0 

-4.5 

-6.0 

-7.0 

4000 

-2.0 

-2.5 

-3.0 

-4.0 

-5.0 

-6.0 

3000 

-2.0 

-2.0 

-2.0 

-2.5 

-3.5 

-4.5 

2000 

-2.0 

-2.0 

-2.0 

-2.0 

-2.0 

-2.0 

1000 

-3.5 

-3.5 

-4.0 

-4.0 

-4.0 

-4.0 

000 

-4.5 

-5.0 

-6.0 

-6.5 

-6.5 

-6.0 

II. 


THE  REVERSED  DUMB-BELL  VORTEX 

v  =  the  observed  tangential  velocities 


215 


10000 

+2.0 

+  2.5 

+  3.0 

+  4.0 

+  4.5 

+  5.0 

9000 

+4.0 

+  6.0 

+  8.0 

.  +  8.5 

+  8.5 

+  9.0 

8000 

+4.0 

+  7.0 

+  10.0 

+  12.0 

+  12.0 

+11.0 

7000 

+5.0 

+10.0 

+  13.0 

+14.0 

+14.0 

+14.0 

6000 

+6.0 

+11.0 

+  14.0 

+15.0 

+15.0 

+16.0 

5000 

+7.0 

+12.0 

+  15.0 

+16.0 

+  18.0 

+19.0 

4000 

+7.0 

+11.0 

+14.0 

+17.0 

+  19.0 

+21.0 

3QOO 

+6.0 

+  10.0 

+13.0 

+16.0 

+19.0 

+23.0 

2000 

+5.0 

+  9.0 

+12.0 

+15.0 

+18.0 

+22.0 

1000 

+5.0 

+  6.0 

+  8.0 

+10.0 

+  12.0 

+14.0 

000 

+5.0 

+  5.5 

+  6.0 

+  6.0 

+  6.5 

+  7.0 

III. 


The  computed  a  A  = 


—  u 


Iff  sin  az      Iff  cos  a  z 


10000 

165 

266 

431 

828 

1593 

6732 

9000 

342 

652 

1152 

1775 

3091 

12577 

8000 

358 

781 

1469 

2523 

4401 

15659 

7000 

467 

1105 

1900 

2998 

5217 

19852 

6000 

537 

1201 

2028 

3180 

5518 

24801 

5000 

595 

1269 

2118 

3270 

6325 

25306 

4000 

585 

1157 

1953 

3439 

6548 

27298 

3000 

506 

1046 

1794 

3187 

6441 

29300 

2000 

431 

946 

1659 

2978 

6040 

27598 

1000 

488 

713 

1220 

2120 

4216 

18198 

000 

538 

762 

1158 

1741 

3064 

11524 

The  unit  =  .00000001  =  1  X  10  -  8. 

It  is  obvious  that  the  velocities  u,  v  can  be  computed  from 
the  formulas,  knowing  the  values  of  the  constants  a  A,  or,  on 
the  other  hand,  the  constants  can  be  computed  from  the  observed 
u,  v  velocities.  The  results  of  this  computation,  Table  55,  Fig. 
27,  show  that  the  dumb-bell  vortex  system  has  been  entirely 
reversed.  The  a  A  lines  are  concave  toward  the  axis,  they  are 
geometrically  spaced  but  closed  up  on  the  outer  rather  than  on 
the  inner  areas  of  the  cyclone.  The  temperature  distribution 
conforms  to  this  arrangement,  and  the  cause  is  probably  due  to 
the  penetration  of  the  vortex  of  the  lower  strata  into  the  rapidly 
moving  drift  of  the  upper  strata. 


216      CONSTRUCTION   OF   VORTICES   IN   EARTH'S   ATMOSPHERE 


Historical  Review  of  the  Three  Leading  Theories  Regarding  the 
Physical  Causes  of  Cyclones  and  Anticyclones  in 
the  Earth's  Atmosphere 

In  the  Astronomical  and  Astro  physical  Journal,  January, 
1894,  the  writer  made  a  summary  of  the  three  most  important 
general  theories  regarding  the  physical  causes  which  generate 
the  local  storms,  called  cyclones,  in  the  earth's  atmosphere,  and 
the  following  extracts  from  that  review  are  sufficiently  explicit 
for  ordinary  purposes.  The  three  theories  are:  (l)  Ferrel's 
warm-center  and  cold-center  cyclones;  (2)  Hands  dynamic 
production  of  temperatures  as  found  by  observation;  (3)  Bige- 
low's  asymmetric  cyclone  with  warm  and  cold  currents  arranged 
in  ridges,  or  streams  of  different  densities,  and  driven  into  local 
cyclonic  and  anticyclonic  circulations  by  the  force  of  gravitation 
acting  upon  them.  Ferrel  had  in  mind  for  his  cyclone  the  type 
of  the  general  circulation  of  the  atmosphere,  and  conceived  that 
the  same  principles  dominate  both  of  them.  The  general  cir- 
culation is  described  as  a  cold-center  cyclone,  with  eastward 
movement  from  the  pole  to  the  high-pressure  belt  in  latitude  33°, 
and  a  warm  ring  of  westward  movement  in  the  tropics;  the  local 
cyclone  is  warm  at  the  center  and  has  right-handed  rotation  from 
the  axis  to  a  ridge  of  high  pressure,  outside  of  which  a  cold  ring 
circulates  in  the  anti-right-handed  direction  in  the  northern 
hemisphere,  these  directions  of  motion  being  reversed  in  the 
southern  hemisphere.  Ferrel's  practical  difficulty  was  to 
account  for  the  originating  heat  energy  in  the  central  column 
of  the  cyclone,  and  this  must  precede  any  criticism  of  the  circula- 
tion that  depends  upon  it.  He  writes,  "In  the  ordinary  cyclonic 
disturbances  of  the  atmosphere,  the  causes  are  similar  to  those 
in  the  general  circulation  but  more  local,  and  consist  of  a  differ- 
ence of  density  arising  mostly  from  a  difference  of  temperature 
between  some  central  area  and  the  external  surrounding  parts 
of  the  atmosphere."  This  dominant  idea  proved  fatal  to  Ferrel's 
successful  development  of  sound  fundamental  principles,  and 
has  greatly  influenced  many  students  to  travel  a  road  whose 


HISTORICAL   REVIEW   OF   LEADING   THEORIES  217 

end  has  never  been  found.  He  was  evidently  unable  to  account 
satisfactorily  for  the  energy  implied  in  the  temperature 
difference  required  to  do  the  work  observed  in  the  motions  of 
the  cyclones  and  anticyclones.  In  his  Coast  and  Geodetic 
Survey  Report,  p.  183,  he  remarks:  "If  for  any  reason  there 
is  kept  up  a  continued  interchange  of  air  between  the  central 
and  exterior  part";  p.  201,  "The  condensation  of  aqueous 
vapor  plays  an  important  part  in  cyclonic  disturbances,  but  is 
by  no  means  either  a  primary  or  a  principal  cause  of  cyclones"; 
p.  239,  "Rainfall  is  not  essential  to  the  formation  of  areas  of 
low  barometer,  and  is  not  the  principal  cause  of  their  formation 
or  of  their  progressive  motion";  in  Waldo's  edition  of  FerrePs 
"Hydrodynamics,"  p.  39,  "The  theory  which  attributes  the 
whole  of  the  barometrical  oscillations  to  the  rarefaction  of  the 
atmosphere  produced  by  the  condensation  of  vapor  in  the  forma- 
tion of  clouds  and  rain  cannot  be  maintained."  However,  being 
hard  pushed  to  find  a  cause  for  his  central  area  of  high  tempera- 
ture in  cyclones,  he  gradually  weakened  from  this  position,  en- 
dorsed Espy's  condensation  theory  of  the  development  of  latent 
heat  by  the  formation  of  clouds  and  rain,  and  in  the  last  year 
of  his  life  could  write  in  Science,  December  19,  1890,  "All  this 
has  been  done  in  the  condensation  theory  of  cyclones,  with 
results  so  satisfactory  as  scarcely  to  leave  a  doubt  as  to  the 
truth  of  the  whole  theory."  This  was  written  in  reply  to  Dr. 
Hann's  revolt  against  'the  sufficiency  of  this  cause  to  produce 
cyclones  as  observed,  who  took  the  ground  that  these  local 
gyrations  are  only  subordinate  whirls  in  the  general  circulation, 
which  depend  upon  the  effects  of  the  equatorial  radiation  only, 
and  are  independent  of  any  local  cause.  Hann  even  went  so 
far  as  to  conclude,  that  "the  actual  motion  of  the  atmosphere  is 
not  a  product  of  the  temperature  (FerrePs  idea),  but  is  in  spite 
of  it;  the  temperature  is  a  product  of  the  motion,"  Science 
May  30,  1890.  Ferrel  was  loyal  to  the  theory  that  temperature 
differences  cause  the  motion  always  and  everywhere,  and  Hann, 
in  adopting  the  inverse  proposition,  has  surely  erred  against  first 
principles. 

FerrePs  mechanical  theory  urged  him  to  adopt  a  ridge  of 


218      CONSTRUCTION   OF  VORTICES   IN  EARTH'S  ATMOSPHERE 

high  pressure  surrounding  every  cyclone,  as  indicated  in  formulas 
(471)  to  (478),  but  this  is  opposed  by  several  fundamental  con- 
ditions which  show  that  it  does  not  conform  with  modern  observa- 
tions. Another  solution  for  the  warm-center  cyclone  is  given  in 
equations  (479)  to  (490),  but  this  is  equally  opposed  by  the 
observed  conditions.  Both  of  these  mathematical  solutions 
have  been  practically  abandoned,  chiefly  because  there  are,  in 
fact,  no  warm-centered  cyclones  and  no  cold-centered  anti- 
cyclones in  existence.  The  hurricane  probably  has  a  warm- 
centered  system  of  motion,  but  it  is  entirely  different  in  structure. 
When  a  cold  sheet  of  air  overflows  a  warm  sheet,  the  warm  sheet 
flows  outward  radially  from  a  central  point  in  all  directions,  like 
the  spokes  on  a  wheel,  and  this  outward  movement  in  the  high 
cloud  levels  drags  behind  it  the  vortex  tube  described  in  Fig.  26, 
and  Tables  51.  This,  however,  is  entirely  different  from  the 
temperature  conditions  of  motion  in  cyclones  and  anticyclones. 
Bigelow  writes  in  the  same  paper  of  the  Astronomical  and 
Astrophysical  Journal,  "I  must  admit  freely  that  I  am  unable 
to  see  in  the  daily  weather  maps  that  formation  as  fundamental 
which  Ferrel  and  meteorologists  generally  assume  to  be  the  pri- 
mary state.  I  propose  to  see  in  temperature  differences,  ar- 
ranged in  waves  or  ridges,  the  true  cause  of  the  observed  pressures 
and  the  antecedent  of  the  precipitation.  It  is  therefore  necessary 
to  account  for  cold  and  warm  temperature  waves  passing  over 
the  United  States."  "The  passage  of  winds  past  each  other  in 
opposite  directions  tends  toward  local  gyrations,  which  all  drift 
eastward  with  the  prevailing  component  in  middle  latitudes. 
All  this  simply  depends  upon  the  difference  between  the  polar 
and  the  equatorial  temperature,  and  is  fully  in  accordance  with 
the  views  of  Ferrel  and  the  latest  expressions  by  Dr.  Hann." 
"The  formation  of  these  low-  and  high-pressure  areas  is  the 
result  of  the  existence  of  the  warm  or  cold  sections  of  waves 
lying  athwart  the  maximum  crest.  From  first  principles  the 
warm  and  cold  masses  will  be  impelled  toward  each  other, 
because  of  the  action  of  gravitation  on  media  of  differing  density. 
They  will  tend  to  encounter  along  or  near  the  ridge  of  greatest 
temperature  variation.  Along  the  line  of  greatest  temperature 


HISTORICAL   REVIEW   OF    LEADING   THEORIES  219 

change,  with  cold  air  to  the  west  and  warm  air  to  the  east  of 
it,  the  gyrating  cyclone  is  formed,  the  couple  existing  from 
the  system  of  causes  thus  described.  Likewise,  along  the  next 
ridge,  with  cold  air  to  the  east  and  warm  air  to  the  west,  and 
often  to  the  south  of  the  maximum  crest,  the  anticyclone  is 
produced.  A  corollary  remark  is  that  the  storm  track  along 
the  north  United- States  seems  to  be  the  effort  of  the  general 
circulation  to  restore  the  permanent  polar  low-pressure  belt 
which  is  interrupted  by  the  continent.  Another  is  that  tornadoes 
and  hurricanes  are  due  to  precisely  the  same  cause,  namely, 
the  juxtaposition  of  masses  of  air  having  great  temperature 
differences."  The  ideas  were  illustrated  by  the  cyclone  of 
November  16,  17,  1893. 

The  origin  of  these  cold  and  warm  waves,  or  ridges  of  different 
densities,  has  been  discussed  at  great  length  in  the  International 
Cloud  Report,  and  the  streams  from  the  cold  north  and  warm 
south  were  called  "leakage"  currents,  because  these  are  in  fact 
sporadic  offshoots  from  the  general  circulation  into  middle  lati- 
tudes. The  warm  currents  in  the  United  States  are  thrown  off 
by  the  Atlantic  center  of  action,  from  the  Gulf  of  Mexico  to  the 
north;  those  upon  southeastern  Asia  from  the  Pacific  center 
of  action ;  those  upon  northwestern  United  States  from  the  same 
Pacific  center  of  action,  or  else  from  the  Arctic  zone  over  British 
America;  those  upon  northwest  Europe,  from  the  Atlantic  center 
of  action,  or  else  from  the  Arctic  circulation.  The  continents 
and  the  oceans  react  upon  the  general  circulation  in  such  a 
manner  as  greatly  to  disturb  and  distort  its  free  operation,  so 
that  finally  southerly  currents  prevail  in  certain  regions  and 
northerly  currents  are  dominant  in  other  regions.  In  the 
United  States  the  southerly  warm  currents  and  the  northerly 
cold  currents  encounter  in  long  streams,  flowing  past  each  other 
in  waves  or  ridges  of  density,  and  under  the  force  of  gravitation 
they  are  compelled  to  flow  in  cyclonic  and  anticyclonic  circula- 
tions toward  a  thermal  equilibrium.  The  exact  mathematical 
conditions  prevailing  at  every  point  have  been  indicated  in 
Chapter  II  of  this  Treatise,  and  in  that  place,  and  in  Bulletin 
No.  3,  Argentine  Meteorological  Office,  1912,  the  practical  details 


220      CONSTRUCTION  OF  VORTICES   IN  EARTH'S  ATMOSPHERE 

have  been  illustrated.  Bigelow,  in  1894,  laid  down  the  principle 
of  the  force  of  gravitation  acting  upon  density  masses  in  alternate 
juxtaposition,  whether  side  by  side  as  in  cyclones,  anticyclones, 
and  tornadoes,  or  in  vertical  superposition  as  in  hurricanes, 
with  intermediate  cases  between  these  principal  positions;  and 
in  1912  he  worked  out  a  method  of  computing  the  very  important 
terms  of  the  radiation  energy  in  the  general  equations  of  motion, 
which  had  heretofore  been  entirely  omitted  from  the  discussions. 
A  very  brief  summary  of  some  of  the  most  important  features 
of  the  general  and  the  local  circulations  are  added  in  this  place, 
though  the  student  must  consult  the  weather  maps  of  various 
countries  for  any  complete  knowledge  of  such  a  complex  subject 
as  the  actual  circulation. 

The  General  and  Local  Components  of  the  Velocities,  Pressures, 
and  Temperatures  in  the  Circulation  of  the  Atmosphere 

The  General  and  the  Local  Components 

There  are  certain  distributions  of  the  temperature,  pressure, 
wind  direction,  and  velocity,  which  are  characteristic  of  the 
general  motions  of  the  atmosphere,  and  others  which  belong  to 
the  local  circulations  peculiar  to  the  cyclones  and  anticyclones. 
It  is  necessary  to  separate  them  from  the  observed  values  that 
are  the  resultants  of  those  two  components.  It  is  practicable 
to  observe  certain  values  of  the  velocities,  pressures,  and  tem- 
peratures at  a  given  station,  from  which  the  general  or  normal 
values  are  computed,  so  that  by  subtracting  in  the  form  of 
vectors  the  normal  values  from  the  observed  values,  the  local  or 
component  terms  may  be  found.  This  great  labor  has  been 
performed  for  the  United  States  and  the  West  Indies,  the  results 
for  the  velocities  being  recorded  in  the  Cloud  Report,  1898,  for 
the  pressures  in  the  Barometry  Report,  1901,  and  for  the  tem- 
peratures in  the  Report  on  Homogeneous  Normals,  1909, 
together  with  numerous  papers  in  the  Monthly  Weather 
Review.  This  Treatise  is  concerned  with  the  methods  of  com- 
putation and  discussion  appropriate  to  meteorology,  rather 
than  with  statistical  results,  so  that  only  a  brief  summary  of 


NORMAL  AND   LOCAL  VELOCITIES   IN   STORMS 


221 


these  data  can  be  presented  in  this  connection.  Unfortunately, 
the  data  of  meteorology  are  so  bulky  that  it  becomes  very  difficult 
to  do  justice  to  the  subject  within  the  limits  of  a  reasonable 
volume. 

The  Normal  and  Local  Velocities  in  Storms 

In  the  Cloud  Report  are  to  be  found  the  resultant  velocities 
and  directions  of  the  wind  at  the  surface,  in  the  cumulus  levels 
(1,000-2,000  meters),  and  in  the  cirrus  levels  (8,000-10,000 
meters),  for  all  parts  of  cyclones  and  anticyclones,  when  the 
centers  of  these  areas  are  located  in  different  parts  of  the  United 
States,  as  the  Dakotas,  the  Lake  region,  New  England,  Colorado, 
Texas,  and  East  Gulf  States,  respectively.  They  were  obtained 
in  the  several  areas  by  making  a  composite  chart  from  about 


FIG.  28.    Observed  stream  lines  of  air  in  the  cumulus  cloud  level  (2,000) 
over  a  cyclone  whose  center  is  in  the  Lake  Region. 

fifty  charts  for  each  type  of  storms.  For  this  purpose  the  United 
States  was  divided  into  small  areas  by  the  parallels  of  latitude  and 
the  meridians,  the  centers  of  the  fifty  storms  were  made  to  coin- 
cide, and  the  vectors  or  arrows  were  transferred  to  a  common 
chart,  from  which  the  resultant  vector  was  carefully  computed. 
These  charts  are  of  great  theoretical  value  for  the  student,  as 
well  as  of  practical  value  for  the  forecaster,  and  they  should  be 


222      CONSTRUCTION  OF  VORTICES  IN  EARTH'S  ATMOSPHERE 

thoroughly  examined.     Specimens  of  these  charts  are  given  on 
Fig.  28  for  a  cyclone  centered  in  the  lake  region,  transcribing  the 


FIG.  29.    Observed  stream  lines  in  the  cumulus  cloud  level  (2,000)  over  a 
cyclone  whose  center  is  in  the  West  Gulf  States. 


FIG.  30.  Observed  stream  lines  in  the  cirrus  cloud  level  (10,000)  over 
a  hurricane  whose  center  is  in  the  South  Atlantic  States.  These  cur- 
rents all  show  that  there  is  a  U-shaped  formation  in  the  circulation  gen- 
erally opening  to  the  northeastward,  though  it  is  also  found  pointing 
westward  and  southward. 

lower  cloud  level;  on  Fig.  29  for  a  cyclone  in  the  West  Gulf 
States  for  the  cumulus  cloud  level;  and  on  Fig.  30  for  a  hurri- 


NORMAL  AND   LOCAL  VELOCITIES   IN   STORMS  223 

cane  in  the  East  Gulf  States,  upon  the  upper  or  cirrus  cloud 
level.  It  is  seen  in  all  these  cases  that  the  currents  of  air  form 
U-shaped  figures  like  the  isobars  and  isotherms,  in  entering  a 
cyclonic  vortex,  and  that  the  eastward  drift  is  locally  diverted 
into  this  configuration.  There  is  a  bridge  across  the  top  of 
the  U-shaped  vortex,  and  in  the  isobars  a  well-defined  saddle 
is  always  constructed,  where  the  high-pressure  areas  are  tem- 
porarily broken  through  in  the  construction  of  a  vortex  circu- 
lation. 

Fig.  31  gives  a  representation  of  a  typical  circulation  of  air 
in  connection  with  the  isobars  in  three  levels:  sea  level,  3,500-foot 
level,  and  10,000-foot  level,  showing  the  relation  of  the  currents 
to  the  isobars.  The  high-pressure  cusps  tend  to  approach  over 
a  bridge  or  saddle  at  C  S  C,  the  pressure  being  lower  to  the 
north  and  to  the  south  of  it.  The  number  of-  the  closed  isobars 
decreases  with  the  height,  and  it  is  usual  for  them  to  disappear 
at  the  level  of  3,000-4,000  meters,  and  sometimes  even  lower. 
This  is  a  proof  that  the  dumb-bell  vortex  which  dominates  in 
hurricanes  has  almost  entirely  vanished  in  cyclones  except  in 
the  lowest  levels,  the  top  being  entirely  depleted  in  the  higher 
levels.  This  throws  back  the  theory  of  cyclones  into  quite  a 
different  category  of  imperfect  vortices  and,  considering  the 
asymmetrical  distribution  of  the  temperature,  it  is  evident  that 
the  currents  are  due  to  pressure  gradients  in  the  thin  sheets  of 
air  of  different  temperatures  in  the  process  of  mixing  in  the 
middle  latitudes.  The  tendency  for  the  currents  to  divide  into 
two  streams  mutually  underrunning  and  overrunning  each  other 
should  be  carefully  noted. 

In  the  Cloud  Report  are  contained  the  data  from  which  the 
average  heights  were  computed  where  the  several  clouds  are 
formed,  and  the  average  velocities  were  deduced  at  which  they 
move  nearly  eastward,  after  the  cyclonic  and  the  anticyclonic 
components  have  been  eliminated.  Table  56  contains  a  summary 
of  the  velocities  in  high  and  low  areas,  the  northward  and  south- 
ward components  over  high  and  low  areas,  the  northward  and 
southward  components  between  the  centers  in  the  warm  and 
cold  streams,  and  the  seasonal  velocities.  See  Fig.  32. 


224    CONSTRUCTION   OF   VORTICES   IN  EARTH'S   ATMOSPHERE 


The  cloud  forms,  stratus,  cumulus,  strato-cumulus,  alto- 
cumulus, alto-stratus,  cirro-cumulus,  cirro-stratus,  cirrus,  occur 
at  certain  well-defined  heights  on  the  average,  and  it  is  found  by 
observation  that  they  drift  over  the  earth's  surface  at  certain 
average  velocities  as  shown  on  the  diagram.  This  increase  of 
velocity  upward,  from  7  meters  per  second  at  the  surface  to  40 


Typical  abnormal  Isobars,  (Sea  Level) 


FIG.  3 1  a. 

FIG.  3 1  a,  b,  c.  Showing  the  relation  of  the  local  circulation  to  the  typical 
isobars  in  high  or  low  areas  of  pressure.  The  closed  isobars  form  a  rough 
vortex,  which  is  supplied  by  the  two-branched  stream-lines  and  gradually 
dies  out  in  the  higher  levels. 

meters  per  second,  is  called  the  eastward  drift  in  middle  latitudes. 
This  is  the  normal  velocity  component  which  must  be  eliminated 
from  the  observed  component  to  produce  the  local  disturbing 
component  of  velocity  due  to  the  cyclones  and  anticyclones 
proper. 

The  eastward  and  the  westward  drifts,  in  the  middle  latitudes 
and  the  tropics,  respectively,  are  shown  on    Figs.  33,  34,  35. 


Typical  abnormatlsobars,  (8,500  foot) 


FIG.  316. 


Typical-abnormal  Isobars,  (10,000  foot) 

FIG.  3 ic. 


226          CONSTRUCTION  OF  VORTICES  IN  EARTH'S  ATMOSPHERE 


TABLE  56 

SUMMARY  OF  THE  VELOCITIES  OF  THE  MOTIONS  OF  CLOUDS  IN  THE  DIFFERENT 
LEVELS  FOR  THE  MIDDLE  ATLANTIC  STATES.  THE  UPPER  CLOUDS  INCLUDE 
Ci.,  Ci.S.,  Ci.Cu.,  A.S.,  A.Cu.,  WITH  THE  MEAN  HEIGHT  8.4  KILOMETERS, 
AND  THE  LOWER  CLOUDS  INCLUDE  S.Cu.,  Cu.,  S.,  AT  THE  MEAN  HEIGHT  2.4 
KILOMETERS. 

Symbols:  Ci.  =  Cirrus. — S.  =  Stratus. — Cu.  =  Cumulus. — A.  =  Alto. 
L    TOTAL  VELOCITY  IN  HIGHS  AND  Lows  WITHOUT  REGARD  TO  DIRECTIONS 


Clouds 

Ci. 

Ci.S. 

Ci.Cu. 

A.S. 

A.Cu. 

S.Cu. 

Cu. 

S. 

Wind 

Height  in  kilo- 
meters 

9.8 

9.8 

8.1 

5.9 

4.5 

2.5 

1.5 

0.9 

0 

High  Areas 

34  9 

39  1 

33  5 

30  2 

23  5 

23  3 

11  2 

11  4 

4  8 

Per  cent 

Northern 
Southern 
Low  Areas 
Total  motion 

38.3 
30.4 

40.8 
44  6 

42.6 
34.8 

39.8 
42  5 

33.9 
30.5 

39.3 
43  8 

31.1 
24.1 

36.0 
39  4 

26.6 
19.7 

29.2 
32  6 

22.7 
18.5 

28.6 
32  9 

10.9 
10.4 

14.6 
17  4 

12.2 
9.5 

11.1 
13  2 

4.9 

4.8 

5.4 
5.3 

"19" 
15 

Southern 

28.3 

36.3 

34.8 

30.5 

24.4 

21.1 

11.8 

8.6 

5.9 

28 

II.    SOUTHWARD  AND  EASTWARD  COMPONENTS  OF  VELOCITIES  IN  HIGHS  AND  Lows 


Clouds 

Ci. 

Ci.S. 

Ci.Cu. 

A.S. 

A.Cu. 

S.Cu. 

Cu. 

S. 

Wind 

High  Areas 

+    1    Q7 

+  1   ft** 

0  60 

0  37 

0  07 

0  32 

0  13 

1  22 

0  69 

+0             IN 
1  T?           W 

4-33  7 

+32  0 

+32  6 

+27  2 

+22  1 

+16  0 

+  51 

+  58 

+  11 

Low  Areas 

10                  VT 

c    OR 

9  24 

3  00 

4  60 

2  38 

4  00 

—  0  11 

—  1  32 

—  0  40 

1    c>                 VXT 

_i_qq    A 

iqc    q 

+37  2 

+31  3 

+24  3 

•4-24  3 

+11  4 

+  78 

+  15 

III.    MEAN  NORMAL  COMPONENTS  OF  VELOCITY  FOR  THE  UNITED  STATES 


1  6 

3  8 

1  8 

—  2  5 

—  1  2 

—  2  2 

—  0  1 

—  1  3 

—  05 

or    war 

+36  6 

+34  0 

4-34  9 

+29  2 

+23  2 

+20  2 

+  83 

+  68 

+  13 

IV.    COMPONENT  VELOCITIES  IN  SELECTED  AREAS  BETWEEN  HIGH  AND  Low 

CENTERS 


Selected  Areas 

+  0  66 

2  11 

+  4  95 

4-2  79 

+  6  24 

+10  22 

+  6  52 

+  5  25 

+  2  23 

+40  1 

+36  9 

+38  7 

+26  5 

+23  7 

+22  1 

+  96 

+  75 

+  32 

Selected  Areas 

3  75 

3  89 

7  34 

7  47 

—  7  78 

—11  13 

—  8  13 

—  7  97 

—  3  25 

+32  7 

+36  9 

+32  1 

+31  0 

+21  9 

+17  1 

+  65 

+  51 

+  02 

V.    SEASONABLE  VELOCITIES  OF  THE  UPPER  AND  LOWER  CLOUDS 


Clouds 

Upper  clouds     8.4 

Lower  clouds    2.4 

Seasons 

June 

Sept. 

Dec. 

March 

Ann. 

June 

Sept. 

Dec. 

March 

Ann. 

20.0 
17.2 

27.7 
19.6 

High  Areas 
Northern 
Southern 
Low  Areas 
Northern 
Southern 

30.8 
24.2 

39.7 
25.9 

33.7 
35.6 

45.2 
30.9 

37.5 
36.0 

47.4 
33.1 

41.7 
27.7 

37.1 
34.8 

35.5 
29.1 

42.6 
32.4 

12.8 
16.3 

19.2 
12.9 

23.2 
17.8 

32.8 
21.9 

24.9 
20.0 

31.5 

18.7 

21.8 
18.2 

27.5 
16.1 

All  velocities  in  meters  per  second  (1  m.p.s.  =  2.2  miles  per  hour) 


NORMAL  AND   LOCAL  VELOCITIES   IN   STORMS 


227 


The  scale  on  Fig.  35  is  twice  as  great  as  on  Fig.  33  or  Fig.  34.  It 
is  seen  that  at  San  Juan,  and  generally  in  the  West  Indies,  the 
westward  drift  in  the  lower  levels  reverses  into  an  eastward 
drift  in  the  upper  levels,  the  transition  occurring  in  the  A.Cu. 
and  A. St.  levels.  Hence,  the  westward  trade  winds  are  shallow, 


Height 

Velocity  Scale 

Metere 

Feet 

Miles 

m.p.h.     22.2                44.7              67.1                90 

10000 
9000 
8000 
7000 
6000 
5000 
4000 
3000 

32808 

m.p.s.       10                   20                 30                 40 

Ci.CiS 
Ci.Cu 

"A.St. 
A.Cu 

S.Cu 

Cu 

St. 
Wind 

6.21_ 
29527- 

- 

\\ 

6.59- 
26247  - 

4.97  _ 
22966- 

$ 

1 

4.35  _ 

19685- 

16404- 

1 

I 

- 

3.11- 

13123- 

}  J 

2.49  _ 
98431 

1.86  _< 
6562- 

x^ 

/      / 

- 

1000 

1.24- 
3281- 

V 

0.62  _ 

/ 

FIG.  32.     The  total  eastward  velocity  in  high  and  low  areas.     Cloud  heights. 


the  greatest  velocity  westward  being  in  the  S.Cu.  level  (2,000- 
3,000  meters),  and  that  they  give  way  to  the  eastward  drift 
which  prevails  over  the  sub  tropics  in  the  higher  levels.  At 
Key  West,  in  the  midst  of  the  North  American  high-pressure 
belt,  the  eastward  and  the  westward  drift  is  small  in  velocity, 
a  similar  reversal  taking  place  in  the  middle  levels.  In  the  middle 
latitudes  of  the  United  States  the  eastward  drift  prevails  in  all 


228      CONSTRUCTION  OF  VORTICES  IN  EARTH'S  ATMOSPHERE 

levels,  the  velocities  increasing  from  the  surface  upward.  On 
these  diagrams  /  stands  for  January  and  D  for  December,  and 
all  the  intermediate  months  of  the  year  are  given  in  the  line  of 


FIG.  33.     The  eastward  drift  above  Washington,  D.  C.,  for  each  month 

in  the  year. 


SCALE  OF  VELOCITY     ^         20        40       60 

METERS  PER  SECOND 
^  ^D 

Ci,Ci.S,Ci.Cu,.          J                      ^^^, 

0 
D 

A.S,  A.Cu,.                                            ^S1*. 

D 
S.Cu,  Cu,S,.                                      **«  <-*x 

Wind                                                         ^ 

\ 

FIG.  34.     The  eastward  and  the  westward  drift  above  Key  West,  Florida. 


vectors.  In  summer  the  velocities  in  the  tropics  for  the  upper 
levels  are  small  and  disturbed  in  direction,  showing  that  the 
circulation  is  diminished  when  the  sun  is  north  of  the  equator. 
In  all  cases  circulation  depends  upon  contrasts  in  temperature, 
so  that  a  vigorous  circulation  occurs  in  winter  rather  than  in 


ANALYTIC  CONSTRUCTION   OF  RESULTANTS 


229 


summer,  when  the  temperatures  of  the  air  in  the  northern  hemis- 
phere are  much  more  nearly  equal  than  they  are  in  winter. 


SCALE  Of  VELOCITY     V *"   METERS  °ER  SECOND 


FIG.  35.  The  westward  drift  in  the  lower  levels  at  San  Juan,  W.  I.,  revers- 
ing into  the  eastward  drift  in  the  A.Cu.,  A.St.  levels  and  the  eastward  drift 
in  the  upper  levels. 

The  Normal  and  the  Local  Isobars  in  Cyclones  and  Anticyclones 

The  Analytic  Construction  of  the  Resultants 

The  general  theory  of  the  separation  of  components  can  be 
illustrated  by  the  following  figures. 

Draw  circles  about  the  pole  (see  Fig.  36)  representing  baro- 
metric pressures  at  some  level  above  the  surface,  ranging  from 
25.4  inches  near  the  pole  to  27.2  in  latitude  20°.  At  two  points 


230      CONSTRUCTION  OF   VORTICES   IN  EARTH'S  ATMOSPHERE 

superpose  a  series  of  local  circles,  1,  2-8,  representing  a  defect  of 
pressure  at  (1),  and  an  excess  of  pressure  at  (4).  Add  the  respec- 
tive values  together  at  every  point  of  intersection,  and  connect 
up  the  pressures  having  the  same  isobaric  value.  The  resultant 
lines  for  (1)  are  seen  at  (2),  and  for  (4)  they  are  seen  at  (5). 


Latitude  20 c 


FIG.  36.     The  formation  of  cyclones  in  the  general  circulation  about  the 
poles  of  the  earth. 

Putting  (2)  and  (5)  in  continuous  figures  the  resultant  disturbed 
values  are  found  at  (3)  and  (6),  the  ordinary  form  of  the  isobars 
observed  in  the  atmosphere  somewhat  above  the  surface,  as  on 
the  3,500-foot  and  the  10,000-foot  planes.  The  same  facts  can 
be  determined  analytically  as  illustrated  by  Fig.  37. 

Take  the  co-ordinate  systems  as  shown  on  the  diagram. 

Let  R  =  the   radius  of    the    circle,  (a.  b)  the   co-ordinates 


ANALYTIC  CONSTRUCTION  OF  RESULTANTS 


231 


of  the  center,  (x.  y)   the    co-ordinates   of   any  point  on    the 
circle. 

The  general  equation  of  the  circle  is 

(644)  (x  -  a)2  +  (y  -  b)z  =  R*. 
Take  5  =  0  and  transpose  the  terms,  so  that, 

(645)  y2  =  -  xz  +  2ax  +  R2  -  a*. 

The  equation  of  condition  for  the  isobar  which  is  the  resultant 
of  successive  circular  isobars  added  to  successive  straight-line 


-1 


FIG.  37.    The  composition  of  right  lines  and  circles  where  the  gradients 
are  twice  as  great  on  the  lines  as  on  the  circles. 

normal  isobars,  is  that  the  sum  of  certain  pair  numbers  shall  be 
constant  on  the  same  line.  Thus,  A  +  B  =  constant,  where 
A  =  n  x,  some  multiple  of  the  ordinate  x,  and  B  =  the  gradient 
number  on  the  circles.  For  example,  take  the  gradient  on  the 
normal  straight  lines  one-half  that  on  the  normal  circles,  so  that 
n  —  HJ  which  is  about  the  average  in  highly  developed  storms. 
Take  successive  circles,  R  =  6,  5,  4,  3,  2,  whose  gradient  numbers 
are  respectively  £  =  0,  -1,  -2,  -3,  -4.  Take  a  =  6,  A  = 
Y^  x,  A  +  B  =  0  for  the  0-line,  and  n  =  %. 

Similarly,  by  taking  the  proper  groups  of  R,  B,  x  for  the 
—  1,   +1  .  .  .   —2,   +2,  .  .  .  lines  in  low  and  high  areas,  we 


232      CONSTRUCTION  OF  VORTICES   IN  EARTH'S  ATMOSPHERE 

obtain  the  resultants  shown  on  Fig.  37.  Curves  can  be  com- 
pounded analytically  through  trieir  equations,  when  the  equations 
of  the  lines  are  known,  but  as  this  is  not  usually  the  fact  for  the 
lines  representing  meteorological  gradients,  and  lines  of  equal 
meteorological  values,  resort  must  be  had  to  some  graphic  proc- 
ess of  construction. 

Graphic  Construction  of  Resultants 

The  first  step  is  to  determine  the  normal  isobars,  isotherms, 
and  wind  vectors  on  certain  selecte'd  planes.  For  the  purpose 
we  have  chosen,  as  appropriate  to  forecasting  requirements  in 
the  United  States,  the  3,500-foot  and  the  10,000-foot  planes, 
besides  the  usual  sea-level  plane;  and  the  Barometry  Report 
contains  the  normals  of  pressure,  temperature,  and  vapor  pressure 
on  these  three  planes,  while  the  Cloud  Report  contains  the 
normal  vectors  for  the  velocities  of  the  air  motions.  We  have, 
therefore,  to  reduce  the  observed  data  to  these  three  planes,  and 
subtract  graphically  the  normals  from  them  to  obtain  the  local 
disturbing  terms  of  the  given  element.  Thus,  in  Fig.  38,  if  we 
obtained  the  oval  lines  from  observation  and  subtracted  the 
right  lines  from  them,  we  should  recover  the  circles  or  local 
components.  Practically,  lay  down  the  normal  values  of  the 
isobars  or  isotherms  on  a  given  plane,  using  transparent  paper, 
superpose  this  upon  the  chart  of  observed  values  on  that  plane, 
and  draw  the  diagonals  of  the  quadrilateral  figures  that  cover 
the  diagram.  In  this  way  the  charts  of  barometric  pressure 
for  the  year  1903  on  three  planes  (of  which  an  example  is 
given  in  Fig.  38)  have  been  decomposed  into  the  elements  of 
the  general  and  the  local  circulations  from  which  we  can  study 
the  general  circulations  on  the  one  hand,  and  the  local  cir- 
culations on  the  other  hand,  without  confusion. 

The  observed  pressures  at  the  stations  of  the  United  States 
can  be  reduced  to  these  planes  by  means  of  suitable  tables,  so 
that  they  are  easily  embodied  in  a  telegraphic  report  without 
delay  to  the  forecast  message.  This  was  done  for  a  few  weeks 
in  a  preliminary  study.  It  is  to  be  noted  that  the  closed  isobars 
of  the  lower  planes  soon  expand  into  loops  in  the  upper  level. 


115~° 110°          105s 1003          95" 905  85°  80° 


125°        120°      115°     110°     105°    100°      95"      9Q-1      85°      80°       7 


115J  110 J  105 J  100 J 


90 J  85°  80°  75° 


FIG.  38.    The  systems  of  isobars  on  three  planes  for  the 
storm  of  February  27,  1903. 


115°     110°  -.1  105°    100°     90°     90°     85 


,".--  J6°^    70°       65° 


FIG.  39.  The  normal  isobars  (continuous  lines)  and  the  disturbing  local  isobars  (dotted 
lines)  in  the  storm  of  February  27,  1903.  The  normal  lines  were  laid  upon  the  observed  lines 
of  Fig.  38,  and  the  dotted  diagonals  of  Fig.  39  drawn.  There  are  five  closed  isobars  on  the  sea- 
level  plane  and  only  two  closed  isobars  on  the  io,ooo-foot  plane.  The  system  of  high  and  low 
areas  on  the  sea-level  charts  soon  opens  up  into  sinuous  lines  in  the  upper  levels.  A  general 
view  of  this  fact  is  given  on  Fig.  40. 


ISOTHERMS   IN  CYCLONES 


235 


The  Normal  and  the  Local  Isotherms  in  Cyclones  and  Anticyclones 
The  isotherms  as  observed  at  a  given  time  in  the  United 
States  are  separated  by  the  same  process  into  the  two  compo- 
nents, the  normal  and  the  local  disturbance  isotherms.  Lay  a 
chart  containing  the  normal  isotherms  over  the  observed  map 
and  draw  diagonal  lines  connecting  up  points  having  the  same 

Observed  Isobars  Local  Component  Isobars 


Pressure 


Pressure 


Low 


High 


FlG.  40.  Scheme  of  the  distribution  of  the  pressures  in  high  and  in  low  areas, 
in  the  observed  and  in  the  component  isobars,  on  the  levels  up  to  10,000  meters. 

These  isobars  are  somewhat  ideal,  but  they  conform  to  conditions  existing 
up  to  the  top  of  the  local  disturbances  in  the  atmosphere,  that  is  to  the  cirrus 
region.  The  winter  storms  are  cut  off  at  6,000  meters  and  even  lower,  while  the 
summer  storms  can  be  traced  much  higher,  on  account  of  the  relative  retreat  of 
the  low  temperatures  to  the  higher  levels.  The  U-shaped  loops  of  the  high 
areas  open  southward,  and  those  of  the  low  area  open  northward,  so  that  in  the 
upper  levels  there  are  sinuous,  not  closed  isobars.  Progress  in  forecasting  con- 
sists in  studying  these  upper  plane  auxiliary  charts,  in  connection  with  the 
corresponding  sea-level  charts. 


236      CONSTRUCTION  OF   VORTICES  IN  EARTH'S  ATMOSPHERE 


30.30 


30.20 


,29.80  7()'29. 90    30.00     30'10 
Isobars  and  Isotherms  on  the  Weather  Map  for  February  27, 1903 


FIG.  41.    The  weather  map  of  February  27,  1903,  showing  the  observed 
isobars  and  isotherms. 


-10 


-10' 
Temperature  Components  for  February  27, 1903 


FIG.  42.  The  normal  isotherms  (full)  and  the  local  disturbance  isotherms 
(dotted)  which  added  together  produce  the  observed  isotherms  of  Fig.  41, 
Fahrenheit  degrees  of  temperature. 


Height  in 

Meters 

6000 


High 


Low 


High 


5000 


4000 


3000 


2000 


1000 


000 


FIG.  43.  Distribution  of  the  high  and  low  temperatures  in  cyclones  and 
anticyclones  up  to  the  height  of  6,000  meters,  showing  the  tendency  to  divide 
into  two  branches  with  the  maximum  departure  near  the  border  of  the  high- 
and  low-pressure  areas.  Centigrade  degrees  of  temperatures. 


238      CONSTRUCTION  OF  VORTICES  IN  EARTH'S  ATMOSPHERE 

difference  of  temperature,  which  are  the  local  disturbing  iso- 
therms. Such  normal  charts  of  temperature  on  three  planes 
are  found  in  the  Barometry  Report.  On  Fig.  48  the  composite 
of  such  a  disturbance  temperature  system  is  given  for  nine 
cyclones,  similar  to  that  of  February  27,  1903  (Fig.  4l).  Fig.  42 
shows  how  the  disturbance  isotherms  cover  adjacent  high-  and 
low-pressure  areas.  These  cold  and  warm  areas,  as  distinguished 
from  the  normal  temperatures  of  the  season  and  place,  are  what 
accompany  all  anticyclonic  and  cyclonic  disturbances.  The 
wind  currents  are  simply  the  effect  of  the  force  of  gravity  trans- 
porting these  masses  of  air  of  different  temperatures,  so  that  the 
cold  mass  underruns  the  warm  masses,  and  the  warm  mass  over- 
runs the  cold  masses  to  the  right  and  the  left  hand,  on  all  the 
levels  simultaneously  from  the  surface  to  the  top  of  the  disturb- 
ance. The  campaign  of  extending  the  temperature  observations 
into  the  higher  levels  is  going  on  in  different  parts  of  the  world, 
but  definitive  results  have  not  been  reached.  There  is  no  very 
general  and  fixed  system  of  temperature  values  to  be  expected, 
because  the  incessant  circulation,  due  to  the  annual  change  in 
declination  of  the  sun,  prevents  the  atmosphere  from  settling 
down  into  a  simple  thermal  equilibrium.  Fig.  43  gives  an 
example  of  the  distribution  of  the  warm  and  cold  areas  in  cyclones 
and  anticyclones  up  to  6,000  meters.  There  is  a  tendency  for 
the  warm  area  to  divide  into  two  branches  to  the  northward, 
and  for  the  cold  area  to  divide  into  two  branches  to  the 
southward.  The  maximum  departure  of  the  temperature  is 
somewhere  between  the  centers  of  the  low-  and  high-pressure 
areas,  and  it  is  not  distributed  symmetrically  about  the  center 
as  was  assumed  by  Ferrel,  and  by  the  early  German  meteorol- 
ogists, in  the  construction  of  their  theories  of  vortex  motion. 
This  asymmetric  theory  of  vortices  was  first  discussed  in  1894 
and  in  the  Cloud  Report  of  1898,  and  the  defects  of  the  other 
theories  were  pointed  out.  The  pressure  and  temperature  data 
of  observation  were  then  entirely  lacking  in  the  upper  strata,  so 
that  several  years  were  allowed  to  elapse  before  the  subject 
could  be  properly  resumed,  as  was  done  in  1906  in  the  series  of 
papers  on  the  "Thermodynamics  of  the  Atmosphere,"  Monthly 


ISOTHERMS   IN   CYCLONES 


239 


Weather  Review,  and  continued  in  1907,  1908,  in  the  series  of 
papers  on  the  vortices  in  the  atmosphere.     The  labor  of  securing 


Observed  isotherms 


Local  component  isotherms 


Temperature 


Temperature 


High 


Low 


High 


High 


Low 


High. 


000 


FIG.  44.  The  observed  isotherms  and  the  local  disturbing  isotherms  in  high 
and  low  areas  of  pressure  from  the  surface  to  10,000  meters.  There  is  a  ten- 
dency for  the  warm  mass  to  ascend  and  rotate  through  about  one  quadrant, 
changing  the  direction  of  the  horizontal  axis  from  N.E.  to  N.W.,  and  for  the 
cold  mass  to  descend  and  rotate  through  one  quadrant  changing  the  direction 
of  the  axis  from  the  S.E.  to  the  S.W.  The  sinuous  lines  in  the  upper  levels 
deepen  in  the  lower  levels,  chiefly  because  the  rapid  eastward  drift  in  the  upper 
levels,  which  smooths  out  all  kinds  of  disturbances,  relaxes  in  the  lower  levels, 
and  permits  the  disturbance  components  to  dominate  more  fully.  Compare 
these  diagrams  with  the  Figs.  41,  42,  4.8,  and  note  the  position  of  the  line  of  0- 
departure.  Further  observations  will  improve  the  accuracy  of  these  diagrams. 

a  sufficiently  large  amount  of  data  in  the  upper  levels,  by  balloon 
and  kite  ascensions,  in  order  to  eliminate  temporary  local  con- 
ditions and  secure  average  values,  is  so  great  that  many  years 


240      CONSTRUCTION  OF  VORTICES   IN  EARTH'S  ATMOSPHERE 

must  elapse  before  meteorology  will  possess  data  for  computa- 
tions of  precision  in  this  field  of  research.  Meanwhile  it  is  very 
important  for  students  to  have  in  mind  a  picture  of  the  general 
phenomena  so  that  useless  discussions  may  be  avoided.  It  is 
particularly  necessary  that  the  temperature  values  due  to  the 
location  of  the  observatory,  as  a  hill  or  mountain,  should  be 
thoroughly  eliminated,  because  the  temperatures  on  an  elevation 
of  land  are  not  the  same  as  that  in  the  free  air  at  the  same  height. 
Otherwise,  errors  at  the  base  station  would  go  into  the  entire 
series  of  gradients  through  a  mistaken  process  of  computation. 

The  Normal  and  the  Local  Velocity  Vectors  in  Cyclones  and  Anti- 
cyclones 

The  results  of  the  observations  made  on  the  velocities  of  the 
cloud  motions  at  Washington,  D.  C.,  are  embodied  in  Fig.  45 
for  the  anticyclones,  in  Fig.  46  for  the  cyclones,  and  in  Fig.  47 
the  entire  system  of  stream  lines  is  laid  down  as  a  whole.  The 
complete  vector  as  observed  in  different  areas  surrounding  the 
center  is  first  plotted,  and  then  the  component  vector  after  the 
mean  eastward  drift  has  been  eliminated.  The  resultant  vectors 
are  long  in  the  upper  levels  and  short  in  the  lower  levels;  the 
currents  are  slightly  sinuous  in  the  upper  levels,  and  as  they 
escape  from  the  eastward  drift  they  become  more  nearly  cyclonic. 
In  the  component  vectors  there  is  a  maximum  velocity  in  the 
3,000-meter  level,  and  a  decrease  in  velocity  upward  and  down- 
ward. The  cyclonic  disturbances  often  penetrate  to  the  10,000- 
meter  level,  but  in  many  cases  they  cease  before  the  cirrus  level 
is  reached,  and  as  a  practical  matter  they  do  not  retain  an  im- 
portant value  above  4,000-5,000  meters,  because  the  eastward 
drift  there  predominates.  It  should  be  noted  that  in  the  cyclonic 
components  the  U-shaped  opening  at  A  is  in  the  northwest 
quadrant  of  the  upper  levels,  but  in  the  northeast  quadrant  of  the 
lower  levels;  in  the  anticyclonic  components  it  is  in  the  south- 
west quadrant  in  all  levels.  As  can  be  readily  inferred,  it  becomes 
a  very  difficult  task  for  the  meteorologist  to  construct  analytical 
formulas  to  cover  these  complex  curves,  and  to  include  the  press- 
ures, temperatures,  and  vectors  under  the  general  equations 


High  Area  Vectors          Anticyclonic  Components 

76  M  P 


4.66  miles 


0.62  mUea 


FIG.  45.    Mean  vectors  of  velocity  and  direction  in  high  areas. 

I  cm.  =  500  kilometers  for  the  distances, 
i   mm.  =  2   meters  per  second  =  4.48   miles  per  hour  for  the  velocity  vectors. 


Low  Area  Vectors 


Cyclonic  Components 


4.66  milet 


0.62  miles 


Surface 


FIG.  46.     Mean  vectors  of  velocity  and  direction  in  low  areas, 
i  cm.  =  500  kilometers  i    mm.  =  2    meters   per   second  =  4-48  miles 

for  the  distances.  per  hour  for  the  velocity  vectors. 


THE   LAND   CYCLONE 


243 


of  motion  requires  unusual  skill.  Some  years  may,  therefore, 
elapse  before  a  satisfactory  general  theory  can  be  perfected.  In 
what  follows  there  is  only  possible  a  series  of  fragmentary  prop- 
ositions regarding  circulations  in  the  atmosphere. 


Observed  total  wind  vectors 


Local  component  vector 


Velocity 


Velocity 


High 


Low 


High 


Low 


High 


FIG.  47.  Observed  total  and  observed  local  component  wind  vectors  connect- 
ing high  and  low  areas. 

The  Land  Cyclone 

It  has  been  shown  that  the  ocean  cyclones  are  imperfect 
vortices  of  the  dumb-bell-shaped  type,  which  depart  from  the 
nearly  perfect  forms  found  in  hurricanes  and  tornadoes,  through 
the  effect  of  certain  resistances  that  are  represented  by  com- 
ponent reversing  vortices.  These  departures  may  become  very 


244      CONSTRUCTION  OF  VORTICES   IN  EARTH'S  ATMOSPHERE 

irregular,  leaving  only  the  remnants  of  pure  vortex  motion 
from  point  to  point  in  the  cyclone,  in  proportion  as  the  new 
system  of  thermodynamic  forces,  due  to  gravity  acting  on  masses 
of  air  of  different  temperatures,  are  not  symmetrically  distributed 
about  an  axis.  In  the  pure  vortex  motion  of  the  tornado  and 
the  hurricane  the* e  was  no  need  to  consider  specifically  the  action 
of  gravity  on  the  vortex  motion,  because  of  the  symmetrical 
disposition  of  the  air  masses  in  superposed  horizontal  layers. 
In  the  cyclones,  on  the  other  hand,  the  differential  action  of 
gravity  on  adjacent  air  masses  of  different  densities  becomes  the 
primary  consideration,  as  demonstrated  in  Chapter  II,  so  that 
the  vortex  action,  though  still  of  influence,  becomes  of  secondary 
dynamic  value.  The  study  of  temperature  distributions  in 
cyclones  and  anticyclones,  together  with  the  corresponding 
velocity  vectors  and  pressures,  must  be  first  determined  by 
observations  before  the  dynamic  theories  can  be  suitably 
applied.  The  ocean  cyclone  has  been  used  as  a  transition  be- 
tween the  hurricane  and  the  land  cyclone,  in  order  to  bring  out 
the  method  of  the  composition  of  vortices.  In  the  land  cyclone 
the  departures  from  the  perfect  dumb-bell  vortex  are  very  great, 
especially  in  the  upper  strata,  where  the  head  of  the  vortex  is 
depleted  by  its  intrusion  into  the  rapid  eastward  drift  whose 
average  velocities  increase  with  the  height  above  the  surface  of 
the  ground.  This  subject  is  so  very  voluminous  that  only  the 
leading  features  can  be  brought  out  in  this  place. 

The  cloud  observations,  made  by  the  United  States  Weather 
Bureau,  1896-1897,  showed  that  when  the  true  cyclonic  com- 
ponents of  velocity  are  eliminated  from  that  of  the  eastward 
drift,  there  remain  a  cold  current  on  the  western  side  of  a  cyclone 
and  a  warm  current  on  the  eastern  side,  and  that  this  arrange- 
ment persists  in  a  general  way  from  the  ground  up  to  the  cirrus 
levels,  10,000  meters.  The  tangential  velocities  v  are  at  a  maxi- 
mum in  the  strato-cumulus  level,  3,000  meters,  and  they  decrease 
downward  and  upward,  the  lower  part  being  the  truncated 
portion  of  the  vortex,  while  the  longer  upper  part  is  gradually 
destroyed  by  degradation  in  the  eastward  drift.  The  radial 
velocities  seem  to  be  inward  from  top  to  bottom,  taking  the 


THE  LAND  CYCLONE  245 

cyclone  as  a  whole;  or  rather  the  inward  flow  on  the  west  and 
the  outward  flow  on  the  east  side  do  not  appear  to  balance  in 
the  different  levels,  so  that  the  mean  velocity  shall  become 
inward  below  and  outward  above,  as  required  by  the  perfect 
dumb-bell  vortex.  In  this  respect  the  funnel-shaped  vortex, 
with  the  tube  pointing  upward,  was  suggested  as  the  proper 
mode  of  analysis,  but  the  analogy  does  not  hold  in  its  details. 
The  determination  of  these  radial  velocities,  upon  which  so  much 
depends,  in  the  upper  strata  is  really  very  difficult,  and  some 
suitably  located  observatory  might  properly  devote  several  years 
of  observations  to  the  elucidation  of  this  point  with  precision. 
It  has  been  proper  to  make  a  resume  of  the  observations,  so  far 
as  was  required  to  bring  out  the  theory  of  the  subject.  We 
began  with  the  pressures  and  then  took  up  the  temperatures 
and  the  velocities  in  the  levels  up  to  10,000  meters. 

The  land  cyclone  differs  from  the  ocean  cyclone  especially 
in  the  fact  that  it  is  not  so  highly  developed  as  a  dumb-bell- 
shaped  vortex.  The  barometric  pressure  in  the  ocean  cyclone 
sometimes  falls  to  28.00  inches  (711  mm.),  while  in  the  land 
cyclone  it  seldom  falls  below  29.00  inches  (737  mm.).  This 
deficiency  of  the  central  areas  in  the  vortex  tubes  is  due  to  a 
variety  of  causes,  but  the  principal  fact  is  that  the  air  masses 
of  different  temperatures  are  placed  side  by  side  on  the  same 
horizontal  plane  instead  of  being  superposed;  and  the  second 
point  is  that  the  penetration  of  the  head  of  the  vortex  into  the 
eastward  drift  of  the  general  circulation  is  followed  by  its  de- 
pletion, which  is  caused  by  stripping  away  from  the  vortex  of 
fragments  of  the  masses  of  ascending  air.  The  meteorological 
data  that  serve  to  illustrate  these  facts  can  be  briefly  presented. 

A  study  has  been  made  of  the  location  of  the  isobars,  the 
variations  in  the  temperature,  and  the  wind  velocity  and  direction 
in  a  large  typical  land  cyclone,  by  constructing  the  mean  values 
for  a  composite  of  nine  selected  cyclones,  March  16,  1876, 
March  27,  1880,  April  18,  1880,  January  12,  1890,  December  3, 
1891,  November  17,  1892,  April  20,  1893,  January  25,  1895, 
November  22,  1898.  They  were  chosen  such  that  the  cyclonic 
center  occupied  nearly  the  same  place  in  the  United  States, 


246      CONSTRUCTION   OF   VORTICES   IN   EARTH'S   ATMOSPHERE 


namely,  the  lower  Ohio  Valley,  and  they  were  about  equally 
developed  at  the  axis.  On  the  weather  map  the  scale  is  1  mm.  = 
10,000  meters.  The  linear  distances  of  the  radii  to  each  isobar 

TABLE  57 
The  Land  Cyclone 

I.  THE  RADIAL  DIMENSIONS 


(i) 

(2) 

(3) 

(4) 

(5) 

(6) 

B 

760 

755 

750 

745 

740 

735 

25 

1250000 

975000 

733000 

508000 

300000 

110000 

log  ztr 

6.09691 

5.98900 

5.86510 

5.70586 

5.47712 

5.04139 

logp 

0.10791 

0.12390 

0.15924 

0.22874 

0.43573 

II.  A/  VARIATION  OF  THE  TEMPERATURE  FROM  THE  MEAN  DISTRIBUTIONS 


s 

-1.3 

-1.5 

-1.1 

-0.5 

+0.2 

+0.9 

S30E 

+  1.3 

+1.7 

+1.9 

+1.4 

+2.3 

+2.7 

S60E 

+2.9 

+2.8 

+2.9 

+2.9 

+3.0 

+3.2 

E 

+2.1 

•    +2.1 

+2.8 

+2.8 

+2.4 

+2.1 

E30N 

+1.7 

+  1.8 

+1.7 

+  1.7 

+  1-2 

+1.0 

E60N 

+2.2 

+2.3 

+2.1 

+  1.8 

+1.6 

+  1.1 

N 

0.0 

+0.1 

+0.1 

+0.7 

+0.8 

+1.3 

N30W 

-1.1 

-1.2 

-1.0 

-0.4 

+0.4 

+1.3 

N60W 

-3.2 

-3.0 

-2.8 

-2.4 

-1.0 

+0.8 

W 

-6.4 

-5.5 

-4.3 

-2.7 

-0.7 

+1.7 

W30S 

-7.1 

-6.1 

-4.7 

-3.3 

-1.1 

+1.0 

W60S 

-5.6 

-4.6 

-3.3 

-2.1 

-0.4 

+1.1 

III.  WIND  VELOCITY  AND  DIRECTION  WITH  THE  ISOBAR 


S 
S30E 
S60E 

6.4   47° 
6.0   45 
6.7   37 

7.3  44° 
7.6  46 
7.8  41 

10.0  43° 
9.1  46 

8.7  44 

11.9  43° 
10.3  44 
9.4  45 

12.6  44° 
10.8  41 
9.9  39 

12.6  46° 
11.0  38 
10.1  35 

E 
E30N 
E60N 

6.9   40- 
7.9   42 
7.8   29 

7.6  42 
8.1  45 
7.6  35 

7.2  43 
8.5  46 
7.8  41 

6.8  41 
8.2  46 
8.0  44 

7.3  39 
7.9  45 
7.8  43 

8.5  37 
8.2  45 
7.6  40 

'N 

N30W 
N60W 

6.5   54 
6.6   55 
8.1   49 

7.3  54 
8.7  51 
10.2  46 

9.0  54 
10.8  48 
11.0  43 

11.1  53 
11.6  45 
11.0  41 

10.9  50 
10.4  42 
10.4  39 

9.3  47 
9.6  39 
10.2  38 

W 
W30S 
W60S 

8.9   44 
6.9   39 
6.7   51 

10.8  40 
8.4  36 
9.4  51 

11.4  38 
9.8  35 
10.7  52 

11.5  35 
10.8  34 
11.2  50 

11.3  33 
11.1  34 
11.1  45 

11.5  31 
11.2  34 
10.7  39 

THE   LAND   CYCLONE 


247 


were  scaled  in  the  N.W.  -  S.E.  and  S.W.  -  N.E.  directions, 
and  the  means  were  taken  for  the  equivalent  circular  isobars. 
All  the  data  of  Table  57  are  given  in  the  metric  measures.  The 
section  I  contains  the  barometric  pressure  B,  the  circular  radii 


N30  W 


E60°N 


N60°VV 


E30°N 


-7 


W30°S 


S60°E 


-3 


FIG.  48.  Land  cyclone  with  circular  isobars  equivalent  to  the  elongated 
cyclones  of  the  United  States,  with  the  temperature  disturbances  and  the 
wind  vectors,  the  center  being  located  in  the  central  valleys. 

ZD-,  and  the  log  p  =  log  ^  n  .     In  a  pure  vortex  log  p  =  constant, 

but  in  the  land  cyclone  log  p  is  not  a  constant.  Hence  it  follows 
that  the  pure  vortex  laws  no  longer  prevail,  though  their  influence 
continues  to  be  felt. 

On  Fig.  48  the  isobars  are  laid  down,  and  radii  are  drawn  for 
every  30  degrees,  making  points  of  intersection  where  the  com- 
putations can  be  concentrated.  The  isobars  are  spaced  too 
widely  near  the  center  of  the  land  cyclone,  that  is,  the  barometric 
pressure  does  not  fall  near  the  axis  sufficiently  to  conform  to 


248      CONSTRUCTION   OF   VORTICES   IN  EARTH'S   ATMOSPHERE 

the  outer  isobars  from  which  the  vortex  is  to  be  constructed. 
The  temperatures  were  scaled  from  the  weather  maps  at  the  72 
points  of  intersection  just  indicated,  and  the  means  for  each 
point  were  taken  and  plotted  on  diagrams,  one  for  each  radius. 
As  a  matter  of  fact,  the  compilation  was  made  in  the  English 
system,  and  this  involved  132  readings  for  each  of  the  nine 
cyclones.  Similarly,  the  wind  directions  relative  to  the  isobars, 
and  the  velocities  were  measured  at  the  same  points.  The  re- 
sults transformed  to  metric  measures  appear  in  sections  II,  III, 
of  Table  57.  After  the  temperatures  at  the  several  points  had 
been  found,  it  was  necessary  to  subtract  from  them  the  average 
undisturbed  temperature  of  the  region,  that  is,  the  normal  tem- 
perature for  the  average  of  the  dates  of  the  years  in  question.  It 
is  desired  to  know  what  disturbance  of  temperature  accompanies 
the  cyclonic  movements  of  the  air,  as  distinguished  from  the  normal 
temperatures  which  are  due  to  the  general  circulation  taken  by  it- 
self. The  section  II  contains  these  differences,  which  are  also 
plotted  on  Fig.  48,  together  with  the  vectorsof  the  wind  circulation. 
It  shows  that  the  maximum  departure  for  the  cold  area  is 
on  the  S.W.  edge  of  the  cyclone,  and  that  the  maximum  depart- 
ure for  the  warm  area  is  in  the  S.E.  quadrant  generally,  the  line 
of  0-departure  running  nearly  due  north  and  south  through  the 
center  of  the  cyclone.  The  mean  angle  of  the  vector  is  i  =  —  43°, 
though  it  ranges  from  27°  to  54°  in  an  irregular  fashion  from 
one  point  to  another.  It  has  been  shown  that  a  similar  asym- 
metric distribution  of  the  temperature  prevails  within  the  lower 
levels,  being  at  a  maximum  of  departure  in  the  strato-cumulus 
level,  3,000  meters,  and  disappearing  above  in  the  cirrus  level, 
10,000  meters.  When  there  are  masses  of  air  of  different  tem- 
peratures on  the  same  level,  the  densities  are  different,  and  the 
action  of  gravity  is  to  set  up  currents  which  cause  the  cold  cur- 
rents to  underflow  the  warm  currents,  and  the  warm  currents  to 
overflow  the  cold  currents.  The  effort  of  gravity  is  to  restore 
the  isobars  to  a  normal  value  when  they  have  been  disturbed 
by  abnormal  temperature  densities.  The  air  masses  are  trans- 
ported from  the  north  or  from  the  south  into  some  middle 
latitude,  where  this  underflowing  and  overflowing  process  sets 


THE    LAND    CYCLONE 


249 


up  the  cyclonic  and  the  anticyclonic  circulations.  This  prin- 
ciple can  be  illustrated  by  a  vertical  section  running  from  west 
to  east  through  a  series  of  cold  and  warm  masses  of  air.  In  a 
cold  mass  the  isobars  are  concentrated  near  the  surface  and 
opened  in  the  upper  levels;  in  a  warm  mass  the  upper  isobars 
are  concentrated  and  the  lower  are  opened.  When  these  cold 
and  warm  masses  alternate  with  one  another  the  cold  underflows 
in  two  opposite  directions,  and  the  warm  overflows  in  two 
opposite  direcions.  In  effect  in  nature,  the  cold  mass  from  the 


"Warm 


Vertical  Section  West  to  East 
Cold  Warm  Cold 


Warm 


W 


E. 


Horizontal  Section  South  to  North 


FIG.  49.  Model  of  the  action  of  gravity  G  in  forming  streams  of  air  which 
underflow  and  overflow  the  warm  and  cold  masses  on  either  side.  The  isobars 
in  warm  masses  are  relatively  open  below  and  closed  above ;  in  cold  masses  they 
are  relatively  closed  below  and  open  above.  Gravity  tends  to  restore  them  to 
the  same  barometric  levels,  and  the  cyclones  and  anticyclones  are  the  effect  of 
this  process  of  circulation  in  the  impure  vortices  of  the  dumb-bell-shaped  type. 
A  thermodynamic  discussion  of  the  cyclone  and  the  anticyclone  has  been 
given  in  Chapter  II,  showing  the  interplay  of  the  general  forces:  gravity, 
pressure,  circulation,  and  radiation.  We  shall  next  give  a  summary  of  the  data 
for  the  land  cyclone,  which  will  include  the  entire  series  of  terms  in  the  equa- 
tions of  motion,  transformed  to  the  vortex  type,  as  in  Formulas  (561),  and 
their  various  modified  forms. 

north  flows  southward  and  divides,  underflowing  two  warm  masses 
on  either  side,  while  the  warm  mass  flows  northward  and  over- 
flows two  cold  masses  on  either  side.  The  result  of  this  complex 
system  of  currents  is  to  produce  the  cyclones  and  anticyclones, 
and  the  tendency  is  to  approach  a  dumb-bell  vortex,  though  the 
resistance  is  too  great  in  general  to  permit  this  to  be  done. 


250      CONSTRUCTION   OF   VORTICES   IN  EARTH'S   ATMOSPHERE 

Recapitulation  of  the  Formulas  for  the  Dumb-B  ell-Shaped  Vortex, 
(526)-(550),  Fig.  18 

We  resume  the  formulas  for  the  dumb-bell-shaped  vortex 
in  connection  with  the  cylindrical  equations  of  motion,  (526) 
to  (550)  and  (406),  illustrated  by  Figs.  14,  18,  including  all  the 
terms:  inertia,  expansion,  deflection,  friction,  radiation,  circula- 
tion, pressure,  and  gravitation.  If  +  i  is  the  angle  between  the 
tangent  and  the  horizontal  velocity  of  motion,  positive  (+  )  on 
the  outside,  negative  (  —  )  on  the  inside  of  the  circle,  we  have 
az  =  90°  +  i,  the  angle  from  the  radius,  which  in  the  complete 
vortex  passes  from  a  z  =  0°  and  i  =  —  90°  on  the  lower  reference 
plane  for  the  air  inflowing  along  the  radius,  to  a  z  =  90°  at  the 
middle  height  and  i  —  0°  tangential,  to  az  —  180°  and  i  =  +  90° 
on  the  upper  reference  plane  for  air  outflowing  along  the  radius. 
The  intermediate  inflowing  and  outflowing  angles  are  all  de- 
termined by  the  relation  of  the  line  integral  of  the  air  flowing 
radially  to  or  from  the  axis,  and  the  surface  integral  of  the  air 
that  rises  in  the  vortex  from  the  surface  upwards.  For  a  z  =  60°, 
i  =  -  30°  and  for  az  =  120°,  i  =  +  30°.  Hence,  we  have  for 

(646)  Angles,  —  cos  az  =  -f-  sin  i, 

+  sin  a  z  =  +  cos  i, 

u 
—  cot  a  z  =  +  tan  ^  =  —. 

(647)  Velocities,    u  =  —  A  aw  cos  a  z  =  -f  A  a  w  sin  i, 

v  =  A  a  iff  sin  az  =  -j-  -A  a  iff  cos  i, 
w  =  2  A  sinaz  =  -f  2  A  cos  i, 

u 

—  =  —  cot  a  z  =  +  tan  ^, 

u  aw  aw 

—  =  --  —  cot  a  z  =  —  tan  ^, 
w  Z  Z 

2u  2u 

w  =  ---  tan  a  z  =  —  cot  i. 
aw  aw 

(648)  Line  Integral,     2x  w  u  =  —  2  ir  A  a  w2  cos  a  z  = 

2  IT  A  aw2  sin  i. 


(649)    Surface  Integral,  xwiw  =  +  2  n  A  w*  sin  a  z  = 

2  x  Aw2  cos  i. 


RECAPITULATION   OF   FORMULAS 


251 


(650)     Ratio, 


=  -  -  =  —  a  cot  <z  z 


W  TffW 


tan 


u 
a—. 

v 


1     2 

(651)     Tangential  Angle,  tan  i  =  —  . 


2u 


u 
cot  az  =  — . 

v 

Meaning  of  the  Tangential  Angle  i 
The  line  integral  of  velocity  is  the  product  of  the  closed  line, 
circle,  ellipse,  or  any  other  boundary  line,  multiplied  by  the  mean 
velocity  at  right  angles  to  it,  or  more  properly  the  integral  of 
the  mass  velocity  per  unit  length  around  the  boundary, 

ff  —  §  uds 

where  u  is  the  velocity  perpendicular  to  d  s  at  every  point. 
For  a  circular  vortex  symmetrically  disposed  to  an  axis  at  the 
radius  uf  the  velocity  u  is  radial  and  the  same  at  every  point  of 
the  boundary  circle  2  TT  tcr,  so  that  the  line  integral  is, 

S  =  2  TT  iff  u 

It  is  assumed  that  the  inflowing  air  at  the  bottom  of  a  vortex, 
for  example  in  a  tornado,  is  not  congested  and  compressed,  and 
therefore  the  inflowing  mass  must  escape  by  rising  upward 
from  one  level  plane  to  another  in  the  surface  integral  I,  =  J  J  w 
dS. 

The  inflowing  mass  2  iriff  u  escapes  vertically  with  the  velocity 
w  through  the  plane  whose  area  is  irttr2,  so  that  the  surface  integral 
is  simply  Trttr2  w,  not  now  counting 
the  impermeable  bottom,   or   the  ^ 

cylindrical  surface.  The  same  mass 
of  air  entering  the  vortex  ra- 
dially and  horizontally  on  one 
plane  escapes  vertically  on  the 
next  adjacent  horizontal  plane, 
and  it  is  the  vortex-constant  a 
and  the  tangential  angle  i  that 
controls  this  flow,  through  the 
equation, 


w 


r2.  w 


FIG.  50.     The  line  integral  and 
the  surface  integral  in  vortices. 


252      CONSTRUCTION   OF   VORTICES   IN   EARTH'S   ATMOSPHERE 

u       aix  v 

—  =  —  tan  i  =  -  tan  ^ 

w        2  w 

1    Zirlffu          2u         u 
Hence,  tan  *  =  _  .  --  r—    =  -      -  =  -,   and  a  tan  i  =   the 

"2  ' 


ratio  of  the  line  integral  to  the  surface  integral. 

Similarly  from  (523)  to  (525)  for  the  funnel-shaped  vortex, 
and  from  (527)  to  (529)  for  the  dumb-bell-shaped  vortex,  we 
may  summarize, 

Funnel  Vortex  Dumb-bell  Vortex 

,      .     u      1  u 

(652)  -  =  -  .  -  =  —  cot  a  z  =  tan  ^. 

V          Z  V 

u  iff  1  u          -or  -of 

—  =—  —  .-  —  =  —  —a  cot  a  z  —  —  a  tan  ^. 

w  2    z  w  2  2 

v  w  v       tcr 

w  ~     ~  2  '  w  ~  2a' 

2  21 

w  =  —  -  .  z  .u.        w  =  -.  —  -  .  .  u. 
zcr  -ay  a  tan  i 

The  connecting  link  between  these  vortices  becomes, 

(653)  -  --  =  tan*', 

a  z 

the  difference  of  sign  depending  upon  the  fact  that  the  vertical 
axis  was  assumed  in  opposite  directions  in  these  vortices.  The 
tangential  angle  i  varies  from  one  plane  to  another,  —  90°  on 
the  lower  reference  plane,  gradually  changing  to  0°  on  the  middle 
plane  where  there  is  no  inflowing  or  outflowing  air,  then  con- 
tinuing to  +  90°  on  the  upper  reference  plane.  These  are  due 
entirely  to  the  supply  of  air  needed  to  balance  the  inflowing  or 
outflowing  line  integrals  with  the  increasing  or  decreasing  vertical 
surface  integrals  over  the  same  planes  in  succession. 

It  has  been  customary  for  meteorologists  to  explain  these 
tangential  angles  in  cyclones  as  the  effect  of  the  deflecting  or 
the  friction  forces.  Thus,  by  equations  (480)  (481), 

AM  n  _  s* 

(654)  tan  2*1  =  -  =  —  —  j—  ,  for  the  inner  part, 

tan  i  2  =  -  =  —  -T-,  for  the  outer  part, 

but  in  fact  the  theory  is  erroneous.  The  deflecting  forces  de- 
pendent upon  A  are  small,  and  those  depending  on  the  friction 


MEANING   OF   THE    TANGENTIAL    ANGLE   *  253 

k  are  nearly  negligible.  The  inflowing  tangential  angle  in 
cyclones  at  the  surface  is  due  to  the  supply  of  air  necessary  to 
compensate  for  the  rising  air  over  the  entire  surface  of  the 
closed  isobar,  and  therefore  depends  upon  the  integral  of  the 
entire  thermodynamic  system.  In  the  hurricane  the  inflowing 
angle  on  the  ocean  shows  the  amount  of  air  that  is  required  to 
supply  the  mass  of  air  that  is  flowing  away  in  horizontal  radial 
directions  in  the  high  levels,  underneath  the  cold  stratum  that 
has  flowed  as  a  sheet  over  the  tropic  region.  Since  tan  i  = 
—  cot  a  z,  the  vertical  distance  of  the  azimuth  plane  from  the  ref- 
erence planes  can  at  once  be  found  and  thence  the  structure  of  the 
entire  cyclonic  vortex  can  be  deduced  by  the  preceding  methods. 
Since  we  are  not  dealing  with  pure  vortices  in  the  case  of  cyclones, 
these  simple  laws  must  be  modified  from  point  to  point  according 
to  conditions,  and  we  proceed  to  evaluate  the  land  cyclone  in 
the  several  complete  terms  of  the  equations  of  motion,  applied 
to  the  dumb-bell  system.  Making  the  substitutions  indicated, 
we  find  the  following  system  of  equations  for  a  symmetrical 
circular  vortex  with  (406)  in  cylindrical  co-ordinates. 

3P       du          du          du      v2 
(655)    Rad1al)-^  =  r,+  «^+Wg-0--- 

2  w3  cos  6  .  v  +  ku+dQ&. 
=  —  +  A2a2tff  —  2  o>3  cos  0  .A  atxrcosi  + 


dv         8  v          80       uv 
Tangential,  0=-+ 


2o>3  cos  0  .  u  +  k  v  + 


0  =  r-T  +  .  .  .  +  2  us  cos  B.A  a  -&  sin  i  + 
ot 


dw         dw          dw  , 

Vertical,  -  —  =  -^  +  u    -  +  w        +  g+  kw+dQz. 


--  ^-  =  -  -  —  4  A2  a  sin  i  cos  i  +g+k  w+d  Qz 
poz      o  t 


254      CONSTRUCTION   OF   VORTICES    IN   EARTH'S   ATMOSPHERE 

These  equations  contain  the  inertia,  convergence,  centrifugal, 
deflecting,  friction,  and  radiation  terms  in  succession,  reduced 
to  a  form  for  computation  at  any  point  where  the  data  are  known. 

Example  of  the  Evaluation  of  the  Terms  in  the  Equations  of  Motion 

for  a  Cyclone 

As  an  example  in  the  evaluation  of  the  several  terms  in  the 
radial,  tangential,  and  vertical  equations  of  motion  for  a  cyclone, 
I  have  taken  the  data  of  Table  57  for  B.  P.  T.  p.  R.  -or.  q.  -  i}  as 
in  Table  58A,  and  have  computed  a,  a  ttr,  A ,  u,  v,  w,  A  iff,  A  P, 
pm,  in  succession.  In  Table  58s  is  a  summary  of  the  several 
terms,  as  indicated  in  the  equations  of  motion,  for  the  pressure, 
convergence,  deflection,  inertia,  and  friction  combined  with 
radiation.  In  the  vertical  component  the  data  and  results  are 
from  Table  25,  (Qi  -  Qo)  in  the  first  column  for  the  levels  000 
to  500  meters,  these  data  being  divided  by  500  to  give  the  heat 
losses  per  meter  in  a  vertical  direction.  The  other  data  of 
Tables  58B  are  reduced  to  the  unit  length  in  all  cases  for  compari- 
sons. The  pressure  term  radially  is  very  much  larger  than  the  sum 
of  the  convergence,  deflection,  and  inertia  terms,  so  that  the 
remainder  friction  plus  radiation  amounts  on  the  average  to 
.0014582  per  meter  in  mechanical  units  (M.  K.  S.)  Since  press- 
ure acts  inward  these  two  terms  act  outward  along  the  positive 
radius.  The  result  shows  how  impossible  it  is  to  balance  the 
terms  of  the  equation  of  motion  without  friction  and  radiation. 
It  is  not  now  known  how  to  separate  friction  energy  from  radia- 
tion energy  directly,  and  thus  evaluate  them  separately.  These 
terms  in  the  tangential  component  are  much  smaller,  and  act 
in  the  antirotational  direction.  The  radial  component  averages 
0.00146,  and  the  tangential  component  is  0.00038,  that  is  about 
one-fourth  the  amount.  It  is  probably  true  that  most  of  the 
tangential  term  is  due  to  friction  alone,  and  since  i  =  about  45° 
we  may  suppose  that  the  same  amount  of  friction  energy  applies 
to  the  radial  component,  leaving  0.00108  for  the  transported  heat 
energy  towards  the  axis  of  the  cyclone.  The  vertical  component 
is  without  friction  and  amounts  to  0.11648,  so  that  more  than 
100  times  as  much  heat  is  transported  upward  vertically  as 
inward  radially. 


EVALUATION  OF  TERMS  IN  EQUATIONS 


255 


TABLE  58A 

SUMMARY  OF  THE  DATA  OF  OBSERVATIONS  FOR  THE  VORTEX  TERMS  IN  A 

TYPICAL  CYCLONE 


Quantities 

(1)      1      (2) 

(3) 

(4) 

(5) 

(6) 

Formulas 

Barom.  pressure,  B 

0.7600 

0.7550 

0.7500 

0.7450 

0.7400 

0.7350 

m—  mercury 

Force  pressure,  P 

101323 

100656 

99988 

99323 

98655 

97990 

(M.K.S.)  system 

Temperature,  T 

290.7 

287.1 

284.9 

283.2 

281.9 

281.1 

South 

281.8 

281.8 

283.1 

283.0 

282.6 

281.7 

East 

262.9 

266.3 

269.2 

272.7 

275.8 

279.0 

North 

266.5 

268.1 

270.1 

273.1 

276.6 

281.2 

West 

Density,  P 

1.2143 

1.2215 

1  .  2227 

1.2219 

1.2193 

1.2145 

South 

1.2526 

1  .  2444 

1  .  2305 

1.2227 

1.2163 

1.2119 

East 

1.3427 

1.3475 

1.2941 

1.2689 

1.2462 

1.2237 

North 

1.3246 

1  .  3080 

1.2897 

1.2613 

1.2427 

1.2140 

West 

Gas  coefficient,  R 

287.033 

287.033 

287.033 

287.033 

287.033 

287  .  033 

Constant 

Radii,   u> 

1250000 

975000 

733000 

508000 

300000 

110000 

Meters 

OA1  K 

180° 

180° 

Vortex  constant,  a 

.  UlO 

= 

12000 

"  9000  +  3000 

45    truncated 

Vortex  product,  a  w 

18750 

14625 

10995 

7620       4500 

1650 

Tube  constants,  A 

.  000341 

.000499  .000910  .001562  .002802 

.  007633 

g    £  _     u  cosec  i 

a  co 

357 

519         655 

986 

1621 

5156 

E 

347 

500 

829 

1457 

2422 

5635 

^  ^  v  see  i 

a  to 

475 

738 

1037 

1510 

2513 

6966 

W 

Velocities,  Q 

6.4 

7.3 

10.0 

11.9 

12.6 

12.6 

S  Meters  per  sec. 

6.9 

7.6 

7.2 

6.8 

7.3 

8.5 

E 

6.5 

7.3 

9.0 

11.1 

10.9 

9.3 

N 

8.9 

10.8 

11.4 

11.6 

11.3 

11.5 

W 

Tangential  angle,  —  i 

47° 

44° 

43° 

43° 

44° 

46° 

S  Inflowing  angle 

40 

42 

43 

41 

39 

37 

E 

54 

54 

54 

53 

50 

47 

N 

44 

40 

38 

35 

33 

31 

W 

Radial  velocity,  u 

-4.68 

-5.07 

-6.82 

-8.12 

-8.76 

-9.06 

S    u  =  q  sin  i 

-4.30 

-5.08 

-4.91 

-4.46 

-4.59 

-5.12 

E 

-5.26 

-5.91 

-7.37 

-8.87 

-8.35 

-6.80 

N       =  A  a  75"  sin  i 

-6.19 

-6.94 

-7.02 

-6.60 

-6.16 

-5.92 

W 

Tangential  velocity,  v 

4.36 

5.25 

7.31 

8.70 

9.06 

8.76 

S    v  =   qcoai 

5.29 

5.65 

5.26 

5.13 

5.67 

6.79 

E 

3.82 

4.29 

5.35 

6.68 

7.09 

6.34 

N       =  A  a  oT  cos  i 

6.40 

8.27 

8.98 

9.41 

9.47 

9.86 

W 

Vertical  velocity,  w 

.000465 

.  000718 

.001458 

.002283 

.004027 

.010618 

s.-lt 

a  <o 

564 

773 

957 

1346 

2520 

8230 

E 

408 

587 

973 

1753 

3116 

7685 

N      =  2  A  cos  * 

683 

1131 

1633 

2470 

4209 

11952 

W     =  —  cot  * 

o5 

Differences  A  w 

275000 

242000 

225000 

208000 

190000 

The  mean  values 

of     two     successive 

Differences  A  P 

667 

668 

665 

668 

665 

tubes   placed    under 

the    second    of    the 

Means  Pm 



1.2179 

1.2221 

1.2223 

1  .  2206 

1.2169 

pair  from  which  they 

1.2485 

1.2375 

1.2266 

1.2195 

1    2141 

are  computed. 

1.3451 

1.3208 

1.2815 

1  .  2576    1   2350 

1.3163 

1.2989 

1.2755 

1.2520 

1.2284 

The  ratio  of  the  line  integral  to  the  surface  integral  checks. 


2u       a  u 

a  tan  *  =  =—  = . 

taw         v 


256      CONSTRUCTION   OF   VORTICES   IN   EARTH'S   ATMOSPHERE 


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£               UQrS 

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3         'i- 
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EVALUATION   OF   TERMS   IN   EQUATIONS 


257 


5l 

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8 

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8 

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t^-         CO 

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t^            rH 

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CO             Oi    rH    rH 
tO             rH    T^    <M 

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O5         CO 

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s  i 

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CO         0 

t>-         O5  CO  (M 

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C^                     flj     QJ 

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3          u. 
C/3        (x< 

258      CONSTRUCTION   OF   VORTICES   IN   EARTH'S   ATMOSPHERE 


8 


colco 


ff 

£ 


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CO  Tt^ 


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(N 


CO 

^.oS 


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tj    ctf    o 
a;  '-M  '^ 


w 


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O5  (N  CO 


IQ 

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II 


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EVALUATION    OF    TERMS   IN   EQUATIONS 


259 


8 


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II « 


58 

H.S 


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260      CONSTRUCTION   OF   VORTICES  IN  EARTH'S  ATMOSPHERE 

Similar  studies  of  the  relation  of  the  heat  and  radiation  terms 
to  the  mechanical  dynamic  terms  may  be  extended  to  the  higher 
levels  of  the  cyclone,  and  to  the  anticyclone,  but  the  final 
conclusion  to  be  remembered  is  that  there  is  no  possibility  of 
balancing  in  a  dynamic  system  of  equations  the  several  terms  of 
motion,  without  including  the  radiation  of  heat  energy,  and  its 
convection  from  point  to  point.  This  branch  of  meteorology 
will  require  much  further  study  along  the  lines  that  have  been 
developed  in  this  Treatise. 


CHAPTER  V 

Radiation,  lonization,  and    Magnetic  Vectors  in  the   Earth's 

Atmosphere 

THE  incoming  solar  radiation  separates  into  two  parts,  the 
first  the  irreversible  heat  that  cannot  be  transformed  back  into 
the  original  energy,  the  second  the  reversible  energy  which 
appears  as  electrical  and  magnetic  forces.  The  heat  energy  is 
observed  as  the  air  temperature  at  different  points,  and  its 
effects  are  found  in  the  general  and  the  local  circulations  of  the 
atmosphere.  The  short  waves  of  the  solar  radiation,  at  very 
high  temperatures,  as  6700°  to  7700°,  are  capable  of  producing 
ions  of  positive  and  negative  electricity  by  the  disintegration  of 
the  atoms  and  molecules  of  the  gases  that  compose  the  air, 
whereby  a  part  of  the  radiation  energy  reappears  by  transforma- 
tion as  free  ions,  or  free  electric  charges,  more  abundantly  in 
some  strata  than  in  others.  These  ions  tend  to  move  in  electric 
streams,  in  certain  general  lines  as  controlled  by  a  series  of  physical 
conditions,  and  in  their  movement  they  induce  magnetic  deflect- 
ing vectors,  which  disturb  the  earth's  normal  magnetic  field, 
through  whose  lines  of  magnetic  force  the  electric  ions  move. 
We  have,  therefore,  to  study  the  distribution  of  the  solar  radia- 
tion in  the  atmosphere,  the  production  of  free  electric  charges 
through  ionization,  and  the  dependent  induced  magnetic  de- 
flecting vectors.  In  spite  of  prolonged  researches  in  these  sub- 
jects by  many  students,  there  is  a  wide  discrepancy  in  the  results, 
as  whether  the  solar  intensity  of  radiation  is  2.00  calories  or  4.00 
calories,  whether  the  absolute  coefficient  of  electric  conduction 
is  2X10~~5  or  6  X10~5,  whether  the  vectors  that  produce  the 
diurnal  variations  of  the  magnetic  field  originate  in  the  higher 
or  the  lower  strata  of  the  atmosphere,  whether  the  sun  is  a  star 
with  a  variable  periodic  output  of  radiation  or  practically  con- 
stant, and  whether  the  annual  changes  of  the  weather  conditions 
are  dependent  upon  solar  variability  or  are  merely  accidental. 

261 


262          RADIATION,   IONIZATION,   AND   MAGNETIC  VECTORS 

The  Determination  of  the  Intensity  of  the  Solar  Radiation  by 
Observations  with  the  Pyrheliometer  and  the  Bolometer 

Measurements  of  the  intensity  of  the  solar  radiation  are  made 
by  the  pyrheliometer,  which  integrates  in  bulk  the  rays  received 
by  the  instrument,  and  by  the  bolometer  which  measures  the 
individual  lines  in  the  energy  spectrum.  The  two  instruments 
supplement  each  other,  because  while  the  pyrheliometer  gives 
no  account  of  the  selective  absorption  of  lines  and  bands  in  the 
spectrum,  the  bolometer  defines  these  depletions,  and  permits 
the  comparison  of  the  observed  spectrum  energy  with  that  of  a 
full  radiator  at  the  given  temperature.  Neither  instrument 
gives  any  account  of  that  portion  of  the  incoming  solar  radiation 
which  is  reflected  back  to  space,  as  the  albedo  of  the  earth,  but 
this  can  be  found  indirectly  by  thermodynamic  computations 
on  the  temperatures  of  the  atmosphere,  as  observed  in  balloon 
ascensions  to  great  elevations. 

The  Pyrheliometer.  This  instrument  consists  of  a  chamber 
for  receiving  a  bundle  of  the  solar  rays,  whose  temperature  can 
be  accurately  measured  at  any  time.  The  temperatures  are 
measured  when  the  solar  rays  are  shaded  by  a  screen,  and  again 
when  exposed  to  the  radiation,  the  sum  of  the  changes  of  tem- 
peratures in  a  given  interval  of  time,  as  one  minute,  being  the 
effect  of  the  radiation  in  temperature  degrees.  A  factor  can  be 
found  by  experiment  which  will  convert  these  temperature 
changes  per  minute  into  calories  per  square  centimeter  per 
minute.  There  are  many  types  of  actinometers  or  pyrheliom- 
eters,  in  which  the  different  materials  used  for  receiving  the 
radiation  are  involved  with  certain  conversion  coefficients. 
The  earliest  form  of  pyrheliometer,  by  Pouillet,  1838,  consisted 
of  a  silver  vessel  rilled  with  a  known  volume  of  water,  the  surface 
being  blackened  to  absorb  all  the  radiation,  whose  increase  in 
temperature  in  a  given  time  could  be  measured  by  a  thermometer 
whose  bulb  was  embedded  in  it.  Silver  box  with  mercury,  copper 
box  with  mercury,  silver  disk  with  no  liquid,  and  many  other 
combinations  have  been  employed.  The  electrical  resistance 
thermometer,  the  Angstrom  double  strip  compensated  pyrheliom- 


DETERMINATION   OF   INTENSITY   OF   SOLAR  RADIATION        263 

eter,  in  which  the  heat  absorbed  is  measured  by  a  compensating 
electrical  resistance,  are  used  with  success.  We  shall  confine 
our  attention  to  Abbot's  silver  disk  pyrheliometer,  which  has 
been  standardized  against  an  elaborate  Primary  Standard 
No.  Ill,  1911,  of  the  Astrophysical  Observatory  of  the  Smith- 
sonian Institution,  and  furnishes  the  comparison  factors  for 
copies  of  the  secondary  pyrheliometers.  This  instrument  leaves 
little  to  be  desired  for  durability  and  accuracy  of  its  operation. 
More  time  is  required  for  the  observations  with  a  thermometer 
system  than  with  an  electrical  resistance  apparatus,  but  the  latter 
needs  much  more  elaborate  auxiliaries,  battery,  galvanometer, 
current,  and  resistance  apparatus,  so  that  it  is  less  readily 
portable,  and  more  liable  to  accidental  inaccuracies  of  adjust- 
ment. 

Theory  of  the  pyrheliometer. 

Let  S  =  the  entire  surface  of  the  body  receiving  radiation. 

5  =  the  cross-section  of  the  rays  falling  upon  it. 

c  =  the  coefficient  of  heat  received  referred  to  water, 


h  =  the  coefficient  of  heat  lost  by  radiation,  dQ2  = 
q  =  the  intensity  of  the  radiation  received  in  dt,  dQi  =  qsdt. 
Hence,  the  general  equation  of  equilibrium  is, 

(656)  dQ  =  dQ1-dQ2  =  qsdt-hSTdt  =  cdT. 

The  shaded  or  cooling  term.  If  the  body  is  in  the  shade, 
g  =  0,  and  we  have, 

7    rr\  1       Ql 

(657)  -=-=  ---dt.      Integrate  for  T=T0,  when  /=0. 

J.  c 

h  S*  h  S* 

(658)  log  T  =  -  -  -  1+  const.  =  -  -  -  1+  log  T0. 

c  c 

T        -*£  *  -*£  t 

(659)  TT   =  e    '<      or     T  =  TQe    ' 

^  0 

This  is  the  cooling  correction  for  T=  the  excess  of  the  tem- 
perature above  the  surrounding  medium,  when  /  is  the  interval 
of  time  elapsed,  and  T0  is  the  initial  excess  at  which  the  cooling 
begins. 


264          RADIATION,    IONIZATION,   AND   MAGNETIC   VECTORS 

The  exposed  or  heating  term.     Divide  the  general  equation  by 
c,  and  we  have, 

(670)  dT=q—dt---Tdt. 

c  c 

From    (656)    the    maximum    temperature  Tm  is    obtained 
when  d  T  =  0, 

(671)  0  =  qs-hSTm,         ~  =  ~  Tm.     Substituting  in  (670), 

(672)  d 


m  . 

C  C  L  m      L  C 

Integrate  for  T  =  T0  when  dt  =  0,  since  dT=-d  (Tm-T), 

(673)  log(r,n-r)=  -  ^l+const.--  --t+log(Tm-T,). 

C  C 

Tm—T  hS  Tm—T         -—  t 

(674)  log  ~^  —  ^r  =  —  -  -  t,  and  -~^  —  ^-  =  e     c     .     Hence, 

1  m  —  1  o  c  l  m  —  l  o 

_h^  t  _hS  t 

(675)  T=T0e    c     +Tm(l-e    c     )  for  the  total  heating  effect. 

This  is  the  equation  for  the  total  effect  of  radiation  when 
exposed  to  the  solar  rays.  It  consists  of  two  parts:  (1)  the 
cooling  term  from  the  initial  temperature  T0}  and  the  heating 
term  reckoned  from  the  possible  maximum  temperature  Tm 
relative  to  that  of  the  surrounding  medium,  where  T  is  the 
effective  difference  of  temperature  at  the  time  /  above  the  sur- 
rounding medium.  It  depends  upon  the  total  surface  S  of  the 
receiving  body,  the  cooling  coefficient  h,  and  the  heating  co- 
efficient of  the  body  c,  referred  to  water. 

An  Example  of  the  Practical  Observations  with  the  Silver  Disk 
Pyrheliometer,  S.  I.  No.  7,  1911 

Several  auxiliary  tables  must  be  prepared. 

1.  The  table  for  the  equation  of  time  for  every  day  of  the 
year. 

2.  The  table  of  the  declination  of  the  sun  for  every  day  of 
the  year. 

3.  The  table  of  2  log  r,  for  the  radius  vector  of  the  earth. 

4.  The  secant  of  the  zenith  distance,  secz,  should  be  com- 
puted through  the  formulas  leading  to  z. 


DETERMINATION   OF   INTENSITY   OF    SOLAR  RADIATION        265 


*o                    <u                   *o                         <u               *o 

III      1    1 

13 

B 

g                            S 

cog      || 

>  m 

4  rH 

rH                                                                   rH 

N                rH  C 

5  rH 

c 

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s         ss       s         ss       3 
r        if  f       r        +  r       r 

^s  K 
< 

ri 

N 

N  00  CO                O  N  IO                lOrHOJ                t-t-CO                OO^ 
NrHrH                   rHOO                   COCON                   OJOO                   •**•**&> 

i  <i 

1      1      1                 If"-*                  1                               +                               III 

gco  oo  w 
COrHO         OOOOOIO         COOOt-00         00  UO  <M  lO         0000«CTj< 

£  O  00 

. 

W§5NN         NtMINN         INNNIM         CJNNN 

to 

OOOO         OOOO         OOOO         OOOO         OOOO 

coco           coco           coco           coco           coco 

c  oo  ^ 

«*, 

I 

"rji                10CO                t-00                00                rHM 

".CO                COCO                COCO                CO1^1                '*^< 

0 

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IN 

266          RADIATION,   IONIZATION,   AND  MAGNETIC  VECTORS 

t      x  tan  5  tan'/  cos  M  tan(<£— M) 

(676)     tan  M  =  — — ,  tan  A  =  ^  (±     1/r, ,  tan  z 


cost 


cos  A 


A  carefully  constructed  diagram  of  curves  is  required  for 
the  interpolations  to  two  decimals  of  sec  z. 

Take  four  readings  of  the  thermometer  every  30  seconds 
for  the  beam  shaded  and  exposed  alternately  during  ten  minutes. 
Compute  the  mean  A7  for  each  group;  take  the  means  by  pairs 
of  the  shaded  terms,  and  subtract  from  mean  exposed  term 
lying  between  them;  the  sum  of  the  two  corrected  A7  gives 
the  rise  in  the  thermometer  degrees  per  minute;  multiply  by 
the  pyrheliometer  factor  0.610  to  obtain  I  in  calories;  take 
log  7  for  the  ordinate  on  the  diagram  of  the  Bouguer  formula. 
Adopt  as  epoch  the  middle  minute  of  the  watch,  running  on 
mean  time,  and  add  the  equation  of  time  +7W  8s  to  reduce  to 
apparent  time;  subtract  this  from  12.00.00  for  the  hour  angle 
t;  take  the  sun's  decimation  of  date  6=+0°  27',  and  with 
/.  6  as  arguments  read  off  sec  z  on  the  scale  diagram  for  La 
Quiaca.  Plot  the  points  (log  7,  sec  z)  on  the  diagram,  and 
draw  the  best  mean  line  through  the  points  of  the  day.  Scale 
off  the  point  (log  70,  sec  2  =  0),  and  reduce  to  the  mean  solar 
distance  by  the  radius  vector  factor  2  log  r  to  obtain  log  70.r 
and  70.r. 


TABLE  60 

LA   QUIACA,    ARGENTINA,    SEPTEMBER  22,    1912 

Similar  pairs  for  September  22,  1912,  at  La  Quiaca  are  as  follows: 


Time 

Log/ 

Sec  z 

Time 

Log/ 

Sec  z 

A.M. 

6.48.00 

0.018 

4.65 

P.M. 

2.0.00 

0.205 

1.24 

Plotted  as  o  for  the  A.M. 

6.58 

0.060 

3.89 

2.10.0 

0.182 

1.33 

Plotted  as  X  for  the  P.M. 

7.8 

0  076 

3.37 

2.20.0 

0.190 

1.38 

8.21 

0.157 

.82 

4.0   .0 

0.141 

2.30 

8.31 

0.180 

.73 

4.10.0 

0.124 

2.52 

8.41 

0.180 

.64 

4.20.0 

0.114 

2.73 

10.28 

0.202 

.17 

10.38 

0.205 

.15 

10.48 

0.202 

.13 

THE  BOUGUER  FORMULA  OF  DEPLETION 


267 


The  most  successful  hours  of  observation  are,  weather  per- 
mitting, 

1.  Group  of  five,  ten  minutes  each,  7.00  to  7.50  A.M. 

2.  Group  of  three,  ten  minutes  each,  8.30  to  9.00  A.M. 


logU 
0.250 

0.200 
0.150 
0.100 

0.050 
0.000 

\ 

loglo 

lo.r 
0.263 

\ 

V 

lofiflo.r 

0.267       1849 

*  \00 

c  IN 

logic 
Ii 

I2 

Is 
-Alogl 
\           log  p 

0.263      for 
.210 
.157 
.104 
-.053 
9.947       0.885 

secz=0 
=1 
=  2 
=  3 

P 

\ 

o^v 

\ 

sec  z 

0.00                1.00                 2.00                 3.00                 4.00                 5.00 

FIG.  51.     Plotting  the  pyrheliometer  observations. 

3.  Group  of  three,  ten  minutes  each,  11.00  to  11.30  A.M. 
The  corresponding  computations  can  be  completed  in  one 
hour,  so  that  three  hours  suffice  to  obtain  I0.r  on  any  good  day. 


The  Bouguer  Formula  of  Depletion 

The  incoming  solar  radiation  is  subjected  to  two  types  of 
depletion,  (1)  the  reflection  and  scattering  of  the  rays,  on  the 
molecules,  ice  crystals,  dust,  and  other  constituents  of  the  at- 
mosphere, whereby  a  certain  amount  of  energy  is  reflected  back 


268          RADIATION,    IONIZATION,    AND   MAGNETIC   VECTORS 

to  space  as  albedo,  and  is  not  subject  to  measurement  by  the 
pyrheliometer.  This  type  of  depletion  affects  chiefly  the  very 
short  waves  in  the  energy  spectrum,  and  its  region  of  operation 
is  especially  in  the  high  cirrus  region;  (2)  the  other  type  of 
depletion  is  due  to  selective  absorption  of  certain  lines  and 
bands  of  the  spectrum,  which  can  be  determined  specifically 
by  bolometer  observations,  with  the  sun  at  different  zenith 
distances  for  the  same  station,  or  by  observations  on  the  radia- 
tion at  the  same  zenith  distance,  from  stations  having  different 
heights  above  the  sea  level.  The  law  of  depletion  is  generally 
expressed  by  Bouguer's  formula, 

(677)  /  =  /0£sec2, 

where  /o  =  the  source  of  radiation  before  depletion, 

^  =  the  fraction  which  is  transmitted  for  the  unit  dis- 

tance, 
sec  z  =  the  secant  of  the  angular  distance  from  the  zenith 

to  the  incoming  ray. 

/  =  the  radiation  measured  at  the  instrument. 
The  formula  is  only  correct  for  homogeneous  rays. 

I.  Passing  to  logarithms. 

(678)  log  /  =  log  70  +  sec  z.  log  p. 

Taking  two  successive  observations  at  Zi,  z2, 

(679)  log  /i  -  log  72  =  (sec  Zi  -  sec  z2)  log  p,  hence, 

(680)  log  p  =  °£   1  ~  °£   2  for  p  constant  in  the  interval. 

sec  zi  -  sec  z2 

II.  If  it  is  assumed  that  sec  z  is  constant  and  p  variable, 
First  line,        log  7  =  log  70  +  log  p.  sec  z. 

Second  line,  log  /'  =  log  /</  +  log  pf.  sec  z. 

(681)  (log  /  -  log  /')  =  (log  70  -  log  7o')  +  (log  p  -  log  p')  sec  z. 

(682)  log  j,  =  log  -p  +  log  j,  sec  z. 


By  the  first  type  of  formula,  it  is  seen  that  log  I  decreases  as 
sec  z  increases.  Taking  (log  /i  —  log  1  2)  for  unit  differences  in 
(sec  Zi  —  sec  z2),  as  in  the  example,  we  easily  find  its  mean  value 


THE  BOUGUER  FORMULA  OF  DEPLETION         269 

—  0.053  for  the  given  line  as  drawn,  by  reading  the  successive 
values  of  log/  as  the  line  crosses  the  sec  z  lines,  0.1.2.3  .  .  . 
whence  p  is  found  from  log  p. 

As  the  aqueous  vapor  rises  from  the  surface  upwards  in  the 
diurnal  convection  the  absorption  (1.00  —  p)  increases,  and  p 
decreases  towards  the  midday,  so  that  the  line  joining  the 
observed  values  of  log  /  is  likely  to  be  a  curve  which  is  slightly 
convex  toward  the  origin.  In  the  dry  climate  at  La  Quiaca 
such  a  curvature  was  not  observed,  though  it  is  common  at  such 
stations  as  Washington  and  Mt.  Wilson.  A  refined  treatment 
of  the  variable  p  can  be  made  for  the  same  zenith  distance  by 
the  second  form  of  the  formula. 

The  Bouguer  formula  indicates  that  the  depletion  of  the 
incoming  radiation  is  proportional  geometrically  to  the  length 
of  the  path  m  which  it  traverses  in  the  atmosphere,  and  it  is 
common  to  the  large  class  of  physical  formulas  which  correspond 
with  similar  conditions.  It  has  been  generally  assumed,  in 
applying  the  formula  to  the  earth's  atmosphere,  that  the  unit 
distance  is  the  depth  of  the  atmosphere  in  the  zenith  mo,  and  that 
what  is  measured  by  the  pyrheliometer  is  the  intensity  of  the 
solar  radiation  at  the  actual  distance  of  the  earth,  before  any 
depletion  takes  place.  It  is  proposed  to  give  several  arguments 
which  show  that  this  is  a  clear  assumption,  which  does  not 
correspond  with  the  facts  of  observation. 

1.  The  Bouguer  formula  contains  two  unknown  terms, 
since  sec  2  is  a  ratio  of  two  distances, 

m        m'       m" 

(684)     sec  s  =  —  =  —  =  —77  =...-• 
mo      mo       mo 

in  which  the  denominator  m0)  m0f,  mo"  is  undetermined;  and 
since  only  log  /  and  log  p  can  be  deduced  from  the  observations, 
log  IQ  is  also  to  be  considered  an  unknown  term.  For  example, 
m0  may  refer  to  one  plane  above  the  observer  as  the  cumulus 
stratum,  m0'  may  refer  to  another  plane  as  the  cirrus  stratum, 
and  m0"  may  refer  to  any  other  plane  as  the  outermost  layer  of 
the  earth's  atmosphere.  The  two  unknown  terms  (log  70,  m0) 
refer  to  the  source  of  the  radiation  7n  on  that  plane  whose  unit 


270          RADIATION,   IONIZATION,   AND   MAGNETIC   VECTORS 

distance  above  the  surface  m0  may  be  any  plane  in  the  atmosphere 
satisfying  these  conditions.  If  the  incoming  radiation  is  partly 
reflected  back  to  space  as  albedo  on  the  cirrus  levels  mQ',  then 


Outer     I( 
layer    — 


Cirrus 
layer 


Cumulus 
layer 


FIG.  52.     Illustrating  the  use  of  the  Bouguer  formula. 

IQ   is  already  diminished  to  that  extent,  and  is  not  subject  to 
observations  by  the  instruments. 

/^o^\  Wl  Wl  Wl 

(685)     sec  zi  =  —  =  —  7  =  —  77  =   .  .  .  . 

mo      mo       m0 

m<t      mz'      m2" 
sec  z2  =  —  =  —  7  =  —  77  =   .  .  .  . 


sec  Zz  =  —  =  —  7 
mo      mo 


m0 


The  Bouguer  formula  contains,  therefore,  a  double  ratio, 

(686)     log  -r  =  —  log  p,  that  is  -7-  and  — 
&/o      m0  IQ        mo 

and  it  is  quite  indeterminate  in  itself,  unless  some  means  can 
be  found  to  fix  the  unit  distance  m0,  whether  70  emanates  from 
the  cumulus,  the  cirrus,  or  the  outermost  layer  of  the  atmos- 
phere. It  has  been  customary  to  make  m0  =  1,  refer  it  to  the 
outermost  layer,  and  thus  assume  that  the  pyrheliometer  meas- 
ures the  so-called  "solar-constant"  /o",  or  intensity  of  the 
sun's  radiation  as  it  falls  upon  the  earth's  atmosphere,  at  W. 


THE   BOUGUER  FORMULA   OF  DEPLETION 


271 


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272 


RADIATION,   IONIZATION,   AND  MAGNETIC   VECTORS 


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II 

274  RADIATION,   IONIZATION,   AND  MAGNETIC  VECTORS 

We  shall  now  give  strong  evidence  for  thinking  that  the  effective 
radiation  Id  is  on  the  lower  levels,  its  value  being  less  than  2.00 
calories,  while  about  2.10  calories  is  reflected  back  to  space  and 
escapes  direct  observation,  so  that  the  true  "solar-constant"  is 
really  more  than  4.00  calories  per  square  centimeter  per 
minute. 

2.  The  depletion  of  the  incoming  radiation  from  a  maximum  value 
on  the  cirrus  levels,  as  determined  by  observations  at  different 
heights. 

Several  series  of  pyrheliometer  observations  have  been  made 
at  stations  having  different  heights  above  the  sea-level,  and  after 
they  have  been  reduced  to  homogeneous  data,  and  referred  to 
a  given  primary  standard,  they  are  comparable,  and  show  an 
increase  in  the  value  of  70.r  with  the  height.  The  primary 
standard  adopted  is  Abbot's  Standard  No.  Ill,  1911,  Smith- 
sonian Astrophysical  Observatory.  The  data  of  Table  13, 
Vol.  II,  Annals,  have  been  reduced  by  the  factor  0.951  and  the 
data  of  the  United  States  Weather  Bureau  (Mt.Weather  Bulletin) 
by  the  factor  1.039,  those  observations  having  been  there  re- 
duced previously  to  the  Angstrom  pyrheliometer  No.  104.  In 
order  to  exhibit  the  average  conditions  that  prevail  at  different 
elevations,  several  10-day  series  have  been  impartially  selected 
from  the  large  amount  of  material  in  hand.  The  individual 
values  (/o-r>  P)  are  given  in  Table  61,  and  in  the  second  section 
their  variations  on  the  10-day  means.  It  is  noted  that  the 
variations  of  the  coefficient  of  transmission  p,  do  not  materially 
differ  with  the  height,  but  there  is  a  change  in  the  mean  values 
of  p,  approximately  0.825  for  all  stations  below  the  1,500-meter 
level,  and  0.900  for  all  stations  above  that  level.  The  variations 
of  A  /o.r  decrease  steadily  with  the  elevation,  and  70.r  increases 
with  the  height  to  a  maximum  value  1.918  which  is  to  be  identi- 
fied as  belonging  approximately  to  the  cirrus  level.  This  is 
explained  more  fully  on  Table  62,  where  the  available  annual 
values  are  collected  for  eight  stations.  Special  attention  is 
directed  to  the  remarkable  results  at  La  Quiaca. 


THE  BOUGUER  FORMULA  OF  DEPLETION 


275 


It  contains  in  section  I,  the  station,  observer,  latitude  <£, 
altitude  in  meters  z,  the  individual  annual  values  for  the  10 
years  1903-1912,  and  the  mean  for  each  station  increasing  from 
1.709  at  Washington  to  1.848  at  La  Quiaca  and  1.861  at  Mt. 
Whitney.  These  means  are  plotted  on  Fig.  53,  as  /o.r,  alongside 
of  p  the  vapor  pressure  of  the  aqueous  vapor  in  grams  per  cubic 
meter  /*,  and  the  relative  absorption  of  A/0.r  by  A/*  =  1.00 
gram  per  500  meters  of  vertical  path  length  as  will  be  explained. 

In  order  to  determine  the  maximum  value  of  70.r  for  the  pyr- 
heliometer  observations,  we  need  to  know  the  bolometer  factor 


Cirrus 

Mt.  Whitney 
La  Quiaca 

Maimara 
Mt.Wilson 
Jujuy 
Mt.  Weather 
Cordoba 
Washington 
Vapor  press 
Radiation  in 
Relative  ab 

10000 

9000 

/* 

[lo.^a 
\        I 

8000 

u 

7000 

st 

I^Sfc, 

6000 

X5- 

^ 

X. 

5000 

\ 

\ 

\ 

("20)     4000- 

- 

\ 

X      \ 

5 

^    3000 

- 

\ 

X^ 

c 

(2384)     2000 

- 

x\, 

^^--•*. 
x^ 

—  C^ 

-^ 

(1780) 
(1302)     1000- 

- 

^1, 

"•-v^ 

N\ 

(526) 

(?I1!  ooo: 

: 

"" 

"\\ 

are                 /J.             0123456789       10 
Calories       Io.r               1.900                 1.850                1.800                1.750                1.700 
sorption          «                 —0.1000         -0.0750          -0.0500            -0.0250          -0.0000 

FIG.  53.     The  relation  between  the  loss  of  radiation  per  i.o  gram  per  cu. 
meter  for  500  meters,  and  the  ratio  between  them,  a. 

F  which  will  supply  the  amount  depleted  by  line  and  band  absorp- 
tion. Mr.  C.  G.  Abbot  has  determined  these  for  Washington, 
(1.123),  and  Mt.  Wilson  (1.094)  from  a  long  series  of  carefully 
executed  bolometer  observations.  These  factors  refer  the 
depletion  to  the  adopted  curve  of  a  perfect  radiator  at  6,000° 
temperature.  At  present,  similar  factors  are  not  available  for 
the  other  stations  in  this  list,  so  that  the  data  of  the  table  in 
reduced  form  are  instructive  rather  than  definitive,  and  it  is 
very  important  to  extend  accurate  bolometer  observations  to 
the  other  stations  employed  in  making  pyrheliometer  observa- 


276          RADIATION,    IONIZATION,    AND   MAGNETIC   VECTORS 


tions.  In  section  II,  the  annual  values  are  all  reduced  to  the 
sea  level  by  the  bolometer  factors  FI,  and  in  section  III  to  the 
cirrus  level  or  approximately  to  the  10,000-meter  plane  by  the 
factors  F2.  The  mean  value  of  I0.r  on  the  sea-level  is  1.711 
calories;  and  on  the  10,000-meter  plane  it  is  1.918  calories. 
Section  IV  contains  the  mean  annual  values  of  p,  approximately 
0.900  above  1,500  meters,  the  top.  of  the  diurnal  convection, 
and  approximately  0.825  below  that  level.  This  difference  in 
the  transmission  factor  p  is  due  to  the  increase  of  density,  p, 
in  the  lower  atmosphere.  An  example  is  taken  from  the  data 
derived  from  balloon  ascensions  for  summer  in  the  temperate 
zone,  which  gives  a  fair  distribution  for  the  year  in  the  low 
latitude  zones. 

TABLE  63 

RELATIVE  EFFICIENCY  OF  1  GRAM  OF  AQUEOUS  VAPOR  PER  CUBIC  METER  IN 
ABSORBING  THE  INCOMING  RADIATION  PER  500  METERS,  MEASURED  IN  CALORIES 


Elevation  in  Meters 

Vapor 

Pressure 

Radiation 

Absorption 

z 

P 

A/z 

I0.r                  l 

^I0.r 

a 

10000 

0.03 

0.02 

1.918       -0 

.002 

-0.1000 

0.05 

0.02 

1.916       - 

.002 

-    .1000 

9000 

0.07 

0.02 

1.914       - 

.002 

-    .1000 

0.09 

0.03 

1.912       - 

.003 

-    .1000 

8000 

0.12 

0.03 

1.909       - 

.003 

-    .1000 

0.15 

0.05 

1.906       - 

.004 

-    .0800 

7000 

0.20 

0.10 

1.902       - 

.006 

-    .0600 

0.30 

0.13 

1.896       - 

.007 

-    .0539 

6000 

0.43 

0.17 

1.889       - 

.007 

-    .0412 

0.60 

0.20 

1.882       - 

.007 

-    .0350 

5000 

0.80 

0.28 

1.875       - 

.009 

-    .0321 

1.08 

0.36 

1.866       - 

.011 

-    .0310 

4000 

1.44 

0.46 

1.855       - 

.015 

-    .0326 

1.90 

0.60 

1.840       - 

.025 

-    .0417 

3000 

2.50 

0.79 

.815       - 

.030 

-    .0380 

3.29 

0.93 

.785       - 

.025 

-    .0269 

2000 

4.22 

1.54 

.760       - 

.016 

-    .0104 

5.76 

1.69 

.744       - 

.014 

-    .0083 

1000 

7.45 

1.91 

.730       - 

014 

-    .0073 

9.36 

0.87 

.716       - 

.006 

-    .0069 

000 

10.23* 

1.710 

/*  =  the  vapor  pressure  in  grams  per  cubic  meter. 
A  /*  =  the  variation  of  n  for  500  meters. 


THE   BOLOMETER   AND   ITS   ENERGY   SPECTRUM  277 

70.r  =  the  observed  radiation  in  calories  per  square  centi- 
meter per  minute,  with  maximum  1.918  on  the  10,000- 
meter  plane. 

A  70.r  =  the  variation  in  the    radiation    or    absorption    per 
500  meters. 

A/o .  r 

(687)  a  =  — —  the  absorption  by  1  gram   per  500  meters. 

The  coefficient  «  multiplied  by  the  observed  change  in  /*  for 
500  meters  gives  the  amount  of  radiation  which  has  been  absorbed 
by  it.  The  curve  of  a  shows  that  there  is  a  gradual  diminution 
of  A  70.r  per  A  /*  =  1  gram  from  10,000  meters  to  the  sea  level, 
with  a  maximum  of  principal  absorption,  chiefly  of  short  wave 
lengths,  in  the  stratum  10,000  to  8,000  meters,  and  a  secondary 
maximum  in  the  stratum  4,000  to  2,000  meters.  The  laws  of 
selective  absorption  in  the  atmosphere  for  different  wave  lengths 
are  very  complex,  and  they  will  require  much  more  research  for 
their  complete  explanation. 

It  is  quite  evident  that  the  curve  of  70.r,  as  derived  from  the 
pyrheliometer  and  bolometer  observations  at  different  elevations 
up  to  4,500  meters,  at  Mt.  Whitney,  cannot  be  extended  much 
above  10,000  meters  without  exaggeration.  Furthermore,  since 
the  aqueous  vapor,  upon  which  the  absorption  chiefly  depends, 
does  not  extend  above  the  10,000-meter,  or  cirrus  levels  generally, 
we  are  compelled  to  place  the  effective  source  of  the  radiation 
observed  by  the  instruments  at  the  cirrus  levels  10,000  to 
20,000  m,  rather  than  on  the  outermost  layer  of  the  atmosphere. 

mf 
Hence,  in  sec  z  =  — -f  the  unit  length  md  is  something  like  13,000 

m$ 

meters,  rather  than  the  full  depth  of  the  atmosphere,  and  the 
depletion,  or  albedo  by  reflection,  is  not  observed. 

3.  The  Bolometer  and  its  Energy  Spectrum  of  Radiation 

The  bolometer  is  a  complex  apparatus,  of  which  the  reader 
can  find  excellent  accounts  in  the  Annals  of  the  Astrophysical 
Observatory  of  the  Smithsonian  Institution.  It  consists  of  a 
siderostat  or  ccelostat  for  directing  a  beam  of  solar  light  in  a 


278          RADIATION,    IONIZATION,    AND   MAGNETIC   VECTORS 

fixed  horizontal  position,  upon  a  slit  which  can  be  adjusted  to 
alter  the  quantity  passing  through  it;  a  converging  and  a 
collimating  mirror  to  focus  the  slit  upon  a  prism,  which  directs 
the  resulting  spectrum  upon  a  reflecting  flat;  a  bolometer  which 
consists  of  a  very  fine  filament  of  platinum,  forming  one  branch 
of  a  delicate  Wheats  tone  balance,  including  a  minute  galvanom- 
eter in  one  branch;  the  lines  of  the  heat  spectrum  falling  upon 
the  bolometer  thread  modify  the  current  of  electricity  in  the 
circuits,  and  deflect  the  galvanometer  which  is  registered  photo- 
'graphically  in  a  manner  to  act  in  synchronism  with  the  position 
of  deviation  of  the  spectrum.  The  movement  of  the  entire 
spectrum  across  the  bolometer  produces  a  spectrum  energy  dia- 
gram according  to  the  prism,  which  must  be  transformed  into 
a  normal  spectrum  of  uniform  dispersion,  and  upon  this  there 
are  very  numerous  line  and  band  deficiencies  of  the  ordinates 
due  to  the  selective  absorptions  that  may  have  occurred.  There 
are  numerous  coefficients  of  absorption  and  reflection  in  the 
mirrors  and  other  parts  of  the  apparatus,  so  that  an  unusual 
degree  of  skill  and  experience  is  required  for  its  successful 
manipulation  and  the  correct  interpretation  of  the  resulting 
ordinates.  This  more  or  less  defective  energy  spectrum  is  to 
be  compared  with  the  full  energy  spectrum  of  a  perfect  radi- 
ator, and  that  of  6,000°  C.  has  been  used  by  the  Smithsonian 
Observatory. 

The  energy  spectrum  of  a  full  radiator  at  a  given  temperature 
T  may  be  computed  by  the  Wien-Planck  formula,  in  which 
/  =  the  energy  of  radiation  at  the  wave-length  X  (x  in  microns, 
V  =  0.001  mm.),  ci  =  575,000,  and  c2  =  14455,  the  adopted 
constants. 

(688)    Wien-Planck  formula,*/  =  a  /~5  (eTr-  l)'1. 

*  Transformations  of  the  (M.  K.  S.)  and  (C.  G.  S.)  systems  for  the 
mechanical  and  heat  units. 

(C.  G.  S.)  System.  1  large  calorie  =  1  Kilogram  degree  1°  C.  water  =  426.8 
Kilogram  meters  =  426.8  X  1000  X  100  X  980.60  =  4.1851  X  1010  ergs 
(Log.  10.62171).  1  small  calorie  =  1  gram  degree  1°  C.  water  =  426.8 
gram  meter  =  426.8  X  100  X  980.60  =  4.1851  X  107  ergs  (Log.  7.62171). 


THE   BOLOMETER   AND   ITS  ENERGY   SPECTRUM  279 

The  total  energy  of  radiation  is  proportional  to  the  fourth 
power  of  the  temperature  and  would  be  expressed  by 

(689)     Stefan  formula, 

/0  =  a  T*  =  —  .  6  —4  T*  Watts  /cm.2  deg.4  (C.  G.  S.). 

iv         c% 

Multiply  by  107  for  ergs/  cm.2  (C.  G.  S.)  Mech.  units,  and 
divide  by  4.1851  X  107  for  gr.  cal./cm.2  sec. 
c\  in  absolute  (C.  G.  S.)  units  is  about  sixty  times  too  large, 
and  depends  on  the  block  (A  A)  included  in  the  spectrum  measures. 
The  constant  is  therefore  more  appropriate  to  a  system  with  the 
minute  for  the  unit  of  time. 

The   displacement  of   the  maximum  ordinate  is  inversely 
proportional  to  the  temperature. 

(M.  K.  S.)  System.  1  large  calorie  =  I  Kilogram  degree  1°  C.  =  426.8  kilo- 
gram meter  =  426.8  X  1  X  9.806  =  4.1851  X  103  (Log.  3.62171). 
1  small  calorie  =  1  gram  degree  1°  C.  =  426.8  gram  meter  =  0.4268  X  1  X 
9.806  =  4.1851  (Log.  0.62171). 

Kilogram       Gr.  X  103        Gr.  N 
Transformalvn  Factor,.     ^^,  =  ^^  =  —,  X  10- 

Mi.KS;  )   X  ^X6°-  .    «!iCaL_  Factor  =0.000014336  (5.15644-10). 

mech.  units/        41851000        cm.2  mm. 

Dimensions  and  transformation  of  the  gravity  equation  from  (M.  K.  S.)  to 

(C.  G.  S.}.    Gravity  equation,   g  fa  -  «„)  =  -    Pl  ~  P°    -  Y^  (tfi2  -  tfo2)  - 

Pio 

(Qi  -  Qo),  (M.  K.  S.).  All  these  terms  as  computed  in  the  (M.  K.  S.) 
system  are  transformed  into  the  (C.  G.  S.)  system  of  mechanical  units  by  the 
Factor  104,  as  can  be  tested  by  substituting  the  terms  dimensionally. 
Similarly  the  equations  (330)  to  (337)  have  the  same  factor  10000.  This 
equation  is  transformed  from  (M.  K.  S.)  mechanical  units  by  the  factor 


1.4336  X  10-5  into  _ 

cm2,  mm, 

Dimensions    and    transformation    of    the    radiation    equation. 

Radiation  equation.      Qi  ~  Qo  -  P10  =  Ul  ~  Uo  =  Klo  =  c  T*. 
v\  —  VQ  Vi  —  VQ 

All  these  terms  as  computed  in  the  (M.  K.  S.)  mechanical  units,  are 
transformed  into   (C.  G.  S.)  mechanical  units  by  the  factor  10,  and  into 

-M*.  by  the  factor      ",*!*      =  0.000014336. 
cm2,   mm.  4.1851  X  107 

The  transformation  of  the  coefficient  in  the  Stefan  Law,  J0  =  a  T*.     The  co- 

<c  •  3  X  1010         6  ci      ,        ci  =  8  TT  c  h  ,      .,     „,      , 

efficient  a  =  7    X   —  p,  where        _     ,  ,,     for  the  Planck  constants 

h  =  6.545  X  10-27,  k  =  1.3606  X  lO"16  (C.  G.  S.). 


280          RADIATION,   IONIZATION,   AND  MAGNETIC  VECTORS 


2891 
(690)     Wien  displacement  formula,  ^max  =  ~^~- 

The  constants  used  in  these  formulas  have  not  been  found 
quite  the  same  at  all  temperatures,  especially  cz  seems  to  have 
increasing  values  for  higher  temperatures,  but  they  serve  suffi- 
ciently for  illustrating  the  principles  now  discussed. 

If  the  sun  is  a  body  which  emits  radiation  at  a  given  tem- 
perature, it  will  have  an  efficient  energy  of  radiation  per  square 
centimeter  per  minute  at  the  surface  of  the  emission. 
Take   R  =  694,800  the  radius  of  the  sun  in  kilometers, 

D  =  149,340,900  the  distance  to  the  earth  in  kilometers 
and  the  effective  radiation  falling  upon  the  earth's  outermost 
layer  of  atmosphere,  before  any  reflection  or  absorption  takes 
place,  is  found  by, 


(691) 


7.90  X  10"11  T4. 


(C.  G.  S.)          ci  =  8  TT  c  h  =  8  X  3.14159  X  3  X  1010  X  6.545  X  10~27  = 

4.93456  X  10-15  (-15.69326) 

C2  =  ch/k  =  3  X   1010  X  6.545  X  10-27/1.3606  x  10-16   = 
•  1.4433  cm.  (0.15928) 
(Mech.  units)    a     =  3.1010  X  6  X  4.93456  X  10-15/4  X  107  X  (1.4433)4  = 

5.1210  X  10-12  (-12.70935) 
(Heat  units)      a     =  5.121  X  10-12  X  107/4.1851  X  107 

1.2236X  10-12  (-12.08764) 

(M.K.S.)        ci    =  8  *  c  h  =  8  X  3.14159  X  3.108  X  6.545  X  IO-34   = 

4.93456  X  IO-24  (-24.69326) 

C2   =  c  h/k  =  3  X  108  X  6.545   X  10-34/1.3606  X  1Q-23  = 

0.014433  (-2.15928) 
(Mech.  units)    <r     =  3  X  108  X  6  X  4.93456  X  10-24/4  X  107  X  (0.014433)4  = 

5.1210  X  10-13  (-13.70935) 
(Heat  units)     a     =  5.121  X  10~13  X  107/4.1851  X  103 

1.2236  X  10-11  (-11.08764) 

In  the  Wien-Planck  Formula  %  in  cm.  becomes  104  ft.     Since  Ci  is  the 
coefficient  of  A5  =  1020  /",  and  c2  that  of  2,  =  104  /*,  we  shall  have, 
(C.  G.  S.)        ci  =  493456     (5.69326),       c2  =  14433  (4.15928). 
(M.K.S.)      cj.  =  4934560  (3.69326),      c2  =  14433  (4.15928). 

These  values  are  computed  for  dry  air.     Kurlbaum's  G  for  ether,  from 
recent  experiments  may  be  taken  provisionally: 


C.  G. 

S. 

Log. 

M.  K. 

S. 

Log. 

a 

in 

cal./sec. 

1.3167 

X 

10-12 

—12.11948 

1 

3167 

X 

io-11 

—11.11948 

a 

in 

cal./min. 

7.900 

X 

io-n 

—11.89763 

7 

.900 

X 

10-io 

—10.89763 

a 

in 

ergs/sec. 

5.510 

X 

io-5 

-  5.74119 

5 

510 

X 

1(H 

-  8.74119 

a 

in 

ergs/min. 

3.3063 

X 

10-3 

—  3.51934 

3 

3063 

X 

10-* 

—  6.51934 

THE   BOLOMETER   AND   ITS   ENERGY   SPECTRUM 


281 


The  following  Table  64  gives  the  solar  energy  of  radiation 
for  certain  selected  temperatures  T.    It  is  computed  with  the 


value  of  ff  =  1.316  X  10  C.  G.  S.,  which  is  not  far  from  the 
value  given  by  Kurlbaum.  In  order  to  make  the  numbers 
comparable  with  the  usual  convention  by  which  the  solar  con- 
stant is  stated  in  equivalent  gram  calories  for  the  time-unit  of 
1  minute,  the  C.  G.  S.  value  of  ff  is  multiplied  by  60  in  (691). 


TABLE  64 

I 

EVALUATION  OF  THE  SOLAR  RADIATION  AT  THE  MEAN  DISTANCE  OF  THE  EARTH 
FOR    SELECTED   TEMPERATURES 


T 

/o 

T 

/o 

T 

/o 

8000° 

7.004 

7000° 

4.106 

6000° 

2.216 

7900 

6.661 

6900 

3.876 

5900 

2.072 

7800 

6.329 

6800 

3.656 

5800 

1.935 

7700 

6.011 

6700 

3.446 

5700 

1.805 

7600 

5.705 

6600 

3.245 

5600 

1.682 

7500 

5.410 

6500 

3.052 

5500 

1.565 

7400 

5.128 

6400 

2.869 

5400 

1.454 

7300 

4.856 

6300 

2.694 

5300 

1.349 

7200 

4.595 

6200 

2.527 

5200 

1.250 

7100 

4.345 

6100 

2.368 

5100 

1.157 

According  to  this  formula  the  equivalent  "solar-constant" 
1.918  would  correspond  with  a  temperature  5,787°.  At  the  time 
the  bolometer  factors  were  determined  it  was  supposed  that  the 
value  IQ.J.  =  2.200  so  that  6,000°  was  assumed  to  be  the  proper 
temperature  for  the  full  radiator,  with  which  to  compare  the 
actual  observed  ordinates  of  the  bolometer  energy  spectrum. 
Successive  improvements  in  the  standardization  of  the  pyrheliom- 
eter  has,  however,  reduced  that  value  to  1.918,  so  that  the 
5,800°  curve  would  be  the  proper  one  to  use  in  determining  the 
bolometer  factor  at  the  several  stations. 

Table  65  contains  the  evaluation  of  the  Wien-Planck  formula 
of  radiation  for  temperatures  from  7,700°  to  5,800°,  together 
with  the  sum  of  the  ordinates  from  /I  =  0.020  p  to  A  =  2.500  p, 
the  maximum  ordinate  Imax,  and  the  maximum  wave-length  Am, 
in  the  displacement  formula.  Abbot  has  determined  the  or- 
dinates in  certain  arbitrary  units  for  a  number  of  wave-lengths, 


282          RADIATION,   IONIZATION,   AND   MAGNETIC   VECTORS 


»S 

a§ 

S^ 

H* 


§       ^ 


<»TO005-*0>^OOCOU300(NOOCO<NlOO>t-r-IOO>lOt-TJ<«OONlO 


0(NOOCO<NlOO>t-r-IOO>lOt-TJ<«OO 

<ot-'^<oaooo«oiot-T-i?OT-(ooiOD3i-io 


NC»U5NC<IC<lTj<O>Tj<lOOeO<'*t-lOOC>eOr-lTl<*O 
«OOiCiOS05CO(Nt>(NO>T--ICOi35NOOCOCOO>OOCSC<I 
iHO>CO«OO»«OO>OC<I«OlO«OO5>OrHO»t-lOTl*eOCO 


THE   BOLOMETER   AND   ITS   ENERGY    SPECTRUM 


283 


and  we  have  finally  decided  to  reduce  them  to  calories  corre- 
sponding approximately  with  6,900°,  and  for  this  purpose  have 
divided  them  by  1,000.  This  will  be  fully  explained. 


T 

6700° 

6600° 

6500° 

6400° 

6300° 

6200° 

6100° 

6000° 

5900° 

5800° 

A.  =0.20|* 

0.779 

0.662 

0.559 

0.470 

0.393 

0.326 

0.270 

0.222 

0.181 

0.146 

.25 

2.209 

1.938 

1.694 

1.474 

1.277 

1.101 

0.945 

0.807 

0.685 

0.579 

.30 

3.743 

3.357 

2.999 

2.671 

2.370 

2.095 

1.844 

1.616 

1.411 

1.226 

.35 

4.844 

4.411 

4.006 

3.626 

3.273 

2.944 

2.639 

2.358 

2.098 

1.859 

.40 

5.382 

4.958 

4.555 

4.177 

3.817 

3.478 

3.160 

2.862 

2.584 

2.322 

.45 

5.458 

5.073 

4.705 

4.353 

4.017 

3.698 

3.395 

3.108 

2.838 

2.583 

.50 

5.232 

4.896 

4.574 

4.264 

3.965 

3.681 

3.408 

3.147 

2.898 

2.661 

.55 

4.841 

4.557 

4.280 

4.014 

3.757 

3.509 

3.270 

3.041 

2.821 

2.610 

.60 

4.378 

4.140 

3.909 

3.683 

3.465 

3.253 

3.049 

2.851 

2.660 

2.477 

.70 

3.451 

3.287 

3.125 

2.968 

2.815 

2.664 

2.517 

2.375 

2.236 

2.101 

.80 

2.662 

2.548 

2.436 

2.326 

2.218 

2.112 

2.009 

1.907 

1.807 

1.710 

.90 

2.046 

1.965 

1.886 

1.808 

1.732 

1.657 

1.583 

1.509 

1.438 

1.367 

.00 

1.577 

1.521 

1.464 

1.409 

1.353 

1.298 

1.245 

1.191 

1.139 

1.089 

.10 

1.226 

1.185 

1.144 

1.103 

1.063 

1.C23 

0.983 

0.944 

0.906 

0.869 

.20 

0.963 

0.932 

0.901 

0.871 

0.840 

0.811 

.782 

.752 

.724 

.695 

.30 

.763 

.740 

.717 

.694 

.672 

.649 

.626 

.604 

.582 

.560 

.40 

.611 

.593 

.576 

.558 

.541 

.523 

.506 

.489 

.472 

.455 

.50 

.495 

.481 

.467 

.453 

.439 

.426 

.413 

.399 

.386 

.372 

1.60 

.403 

.392 

.381 

.371 

.360 

.349 

.338 

.328 

.318 

.307 

1.70 

.332 

.323 

.315 

.306 

.297 

.289 

.280 

.272 

.263 

.255 

1.80 

.276 

.268 

.262 

.255 

.248 

.241 

.234 

.227 

.220 

.214 

1.90 

.231 

.225 

.219 

.214 

.207 

.202 

.197 

.191 

.185 

.180 

2.00 

.196 

.189 

.185 

.180 

.176 

.171 

.166 

.162 

.157 

.152 

2.10 

.165 

.161 

.157 

.153 

.149 

.145 

.141 

.138 

.134 

.130 

2.20 

.140 

.137 

.134 

.131 

.127 

.124 

.121 

.118 

.114 

.111 

2.30 

.121 

.118 

.115 

.113 

.109 

.107 

.104 

.101 

.099 

.096 

2.40 

.104 

.102 

.099 

.097 

.095 

.092 

.090 

.088 

.085 

.083 

2.50 

.090 

.088 

.086 

.084 

.082 

.080 

.078 

.076 

.074 

.072 

Sums. 

52.718 

49.247 

45.950 

42.826 

39.857 

37.038 

34.392 

31  .  873 

29.515 

27.281 

Max. 

5.474 

5.077 

4.704 

4.354 

4.024 

3.714 

3.424 

3.153 

2.899 

2.661 

*  max. 

0.432 

0.438 

0.445 

0.452 

0.459 

0.466 

0.474 

0.482 

0.490 

0.498 

Fig.  54  contains  the  diagrams  of  the  energy  curves  for  7,700°, 
6,900°,  5,800°,  and  Abbot's  observed  ordinates,  which  closely 
agree  with  the  6,900°  curve  from  wave-length  A  =  0.50  /*  to 
A  =  1.00  p.  It  is  seen  that  there  is  a  heavy  depletion  from 
X  =  0.00  to  0.30  {J.  or  0.35  ft,  a  slight  excess  from  A  =  0.50  n  to 
0.70  fi,  a  slight  deficiency  from  A  =  0.70  j«  to  1.00  j«,  and  a  rapid 
increase  in  the  ordinates  to  agree  with  those  of  the  7,700°  curve 
for  A  =  1.50  j«  to  2.50  /*.  Whether  the  matching  of  the  observed 
with  the  computed  ordinates  is  made  for  the  curves  6,500°, 
6,600°,  6,700°,  6,800°,  6,900°,  7,000°,  the  preceding  description  of 
results  holds  true,  but  the  best  coincidence  seems  to  be  for  6,900°. 


284          RADIATION,    IONIZATION,    AND   MAGNETIC   VECTORS 


0.00/X  0.50  M  1.00/A  1.50  jU,  2.00  P  2.50 /*  3.00 /* 


FIG.  54.     Observed    bolometer    ordinates   in    their   relation    to   the    perfect 
radiation  spectrum  at  different  temperatures. 


THE   BOLOMETER   AND   ITS   ENERGY    SPECTRUM  285 

Now  this  corresponds  with  3.876  for  the  solar  constant  at  the 
earth,  instead  of  1.918  as  given  for  the  maximum  from  pyrheli- 

Q   Q7£t 

ometer  observations,  and  this  requires  -      -  =  2.02   times  as 


many  calories  to  fall  upon  the  surface  of  the  outer  layer  of  the 
atmosphere  as  upon  the  pyrheliometers,  so  that  at  least  1.958 
calories  must  have  been  reflected  back  into  space  as  albedo, 
which  would  therefore  be  about  51  per  cent  of  the  radiation  of 
the  sun  received  at  the  earth.  If  the  bolometer  ordinates  can  be 
more  accurately  determined  this  ratio  may  be  definitely  found; 
if  the  ratio  varies  from  year  to  year  at  the  same,  or  at  several 
stations,  the  proper  distribution  between  the  variability  of  the 
solar  radiation  and  the  variability  of  the  terrestrial  reflection 
can  be  further  discussed.  At  present,  one  must  be  very  con- 
servative in  attributing  to  solar  variation  the  entire  apparent 
variations  of  the  radiation  measured  by  the  pyrheliometer  and 
the  bolometer. 

The  remarkable  fact  appears  to  be  established  by  Abbot's 
bolometer  ordinates,  Table  65,  that  these  ordinates  do  not  cor- 
respond with  any  single  temperature  of  emission.  By  comparing 
the  Abbot  ordinate  at  any  given  wave-length  with  the  ordinates 
computed  at  different  temperatures  for  the  same  wave-length,  it 
is  not  difficult  to  interpolate  for  a  temperature  of  emission  that 
would  produce  the  ordinate.  Thus,  they  are  6,850°  at  A  =  0.40  /*, 
6,960°  at  A  =  0.55  /*,  7,260°  at  A  =  1.30  /*,  7,800°  aU  =  1.60  /«  to 
2.00  p.  On  the  face  of  it,  the  Abbot  ordinates  range  through  1  ,000° 
temperature,  6,700°  to  7,700°,  and  this  may  have  several  inter- 
pretations. (1)  The  solar  envelope  may  consist  of  layers  of 
different  temperatures,  7,700°  at  the  photosphere,  which  would 
be  the  general  source  of  emission  for  all  wave-lengths,  gradually 
diminishing  to  6,700°  at  the  top  of  the  chromosphere,  or  possibly 
the  inner  corona,  in  which  envelope  there  is  gradual  selective 
absorption  of  certain  wave  -  lengths,  so  that  the  effective 
emission  of  the  sun  to  space  is  very  complex,  and  corresponds 
with  the  observed  bolometer  spectrum;  (2)  a  similar  selective 
depletion  of  a  uniform  spectrum  of  7,700°-energy  may  occur  in 
the  middle  and  lower  region  of  the  earth's  atmosphere,  so  that 


286          RADIATION,   IONIZATION,   AND  MAGNETIC  VECTORS 


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THE  BOLOMETER  AND  ITS  ENERGY   SPECTRUM 


287 


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288 


RADIATION,   IONIZATION,    AND   MAGNETIC   VECTORS 


the  observed  spectrum  is  only  the  remainder  that  has  escaped 
from  the  terrestrial  depletion  in  the  lower  levels;  (3)  a  mix- 
ture of  these  two  cases  may  be  more  closely  related  to  the  facts, 
but  if  so,  there  will  be  great  difficulty  in  disentangling  the  ele- 
ments. Corresponding  with  these  temperatures  we  may  compute 
the  value  of  the  radiation  at  the  earth,  on  the  first  supposition, 
that  a  depleted  radiation  escapes  from  the  solar  envelope,  and 
we  find  3.765  calories  for  X  =  0.45  /<,  about  4.000  for  A  =  0.50  fJ. 
to  0.60  /*,  a  lower  variable  amount  from  A  =  0.70  /*  to  1.00  /*, 
4.751  at  *  =  1.30  /*,  5.705  at  /l  =  1.50  /£,  6.329  at  A  =  1.60  p, 
6.169  at  A  =  2.00  p.  The  evidences  indicate  that  the  solar 
intensity  of  radiation  at  the  earth  is  from  4.00  to  6.00  calories 
per  square  centimeter  per  minute. 

The  isothermal  region  radiates  2.11  times  as  much  heat  as 
does  the  convectional  region.  The  isothermal  region  is  separated 
from  the  convectional  region  by  a  wedge-shaped  layer,  which 


FIG.  55.     The  courses  of   the  relative  radiation. 

partly  reflects  the  incoming  ray  and  partly  absorbs  it.  The 
remainder  proceeds  through  the  convectional  region,  suffering 
depletion  by  minor  absorptions  to  the  cumulus  layer,  where  it 
is  more  rapidly  absorbed,  and  the  balance  reaches  the  sea  level 


THE   BOLOMETER   AND   ITS   ENERGY    SPECTRUM 


289 


or  land  areas.  The  outgoing  ray  experiences  a  similar  complex 
series  of  absorptions  and  reflections.  The  absorption  is  accom- 
panied by  transformation  of  energy  into  ionization  currents  in 
the  cirrus  layer  and  in  the  cumulus  layer,  the  electric  streams  of 
the  former  flowing  to  the  poles  cause  the  auroras,  and  by  induc- 
tion the  aperiodic  variations  in  the  magnetic  field;  the  ionization 
currents  of  the  cumulus  region  are  controlled  by  the  diurnal 
convection  circulation,  and  induce  the  diurnal  variations  of 
the  magnetic  field. 


TABLE  67 


dQ 


APPROXIMATE  MEAN  VALUES  OF  -~f  IN  THE  RADIATION  EQUATION  — jy-  = 

dz  at 


k2    -    PER  1000  METERS 

a  z 


3 

90° 

80° 

70° 

60° 

50° 

40° 

30° 

20° 

10° 

0° 

19000 

-245 

-250 

-256 

-271 

-270 

-286 

-310 

-325 

-288 

-248 

18 

-245 

-266 

-275 

-275 

-282 

-304 

-354 

-367 

-310 

-241 

17 

-277 

-272 

-286 

-283 

-296 

-302 

-345 

-386 

-308 

-259 

16 

-282 

-289 

-294 

-284 

-309 

-321 

-341 

-357 

-303 

-244 

15000 

-294 

-303 

-309 

-317 

-336 

-314 

-320 

-288 

-254 

-207 

14 

-306 

-321 

-322 

-333 

-327 

-311 

-284 

-196 

-176 

-170 

13 

-322 

-343 

-348 

-361 

-379 

-353 

-233 

-145 

-148 

-142 

12 

-335 

-352 

-384 

-396 

-427 

-361 

-113 

-  87 

-100 

-103 

11 

-352 

-329 

-309 

-200 

-180 

-110 

-  49 

-  40 

-  37 

-  52 

10000 

-262 

-278 

-236 

-182 

-108 

-  25 

-  23 

-  79 

-  59 

-  52 

9 

-167 

-163 

-144 

-  96 

-  58 

-  51 

-  92 

-  85 

-  78 

-  76 

8 

-168 

-165 

-136 

-  78 

-  86 

-151 

-145 

-114 

-113 

-110 

7 

-172 

-152 

-149 

-143 

-121 

-154 

-160 

-149 

-145 

-148 

6 

-192 

-163 

-166 

-146 

-151 

-152 

-156 

-152 

-162 

-178 

5000 

-215 

-180 

-188 

-164 

-163 

-146 

-130 

-157 

-168 

-173 

4 

-226 

-202 

-196 

-134 

-159 

-162 

-131 

-159 

-169 

-164 

3 

-234 

-211 

-216 

-140 

-154 

-170 

-174 

-182 

-171 

-164 

2 

-260 

-239 

-232 

-210 

-189 

-140 

-147 

-126 

-103 

-  78 

1 

-140 

-125 

-126 

-110 

-  92 

-  62 

-  46 

-  35 

-  19 

-  24 

0000 

The  mean  values  of  -r*  are  collected  in  Table  67  and  plotted 

(l  Z 

in  Fig.  55.     The  radiation  increases  downwards  to  a  maximum 
at  the  12,000-meter  level,  then  there  is  a  large  increase  in  the 


290          RADIATION,   IONIZATION,   AND  MAGNETIC  VECTORS 

heat  supply  by  absorption  in  the  cirrus  region,  then  an  increase 
in  radiation  to  the  cumulus  level,  and  a  second  supply  of  heat  is 
at  hand  extending  to  the  surface. 

d2  0 

The  Values  of  -y¥  in  the  Radiation  Equation 
J  dz2 

4.  There  is  another  important  argument  leading  to  the  same 
conclusion,  that  the  so-called  " solar-constant"  exceeds  4.00 
calories,  derived  from  the  thermodynamic  computations  described 
in  Chapter  III,  the  results  of  the  same  being  more  fully  published 
in  Bulletin  No.  3,  Oficina  Meteorologica  Argentina,  1912. 
The  value  of  (Qi  —  Q0),  the  loss  of  heat  by  radiation  for  every 
1,000  meters  difference  of  level,  and  for  every  10  degrees  of 
latitude  from  the  equator  to  the  pole,  was  derived  from  the 
temperature  data  of  the  balloon  ascensions.  Table  66  contains 

/d2O 
-7-7,  which  are  found  in  the  radiation 

formula, 

r692)  *Q__  k2<?Q 

(692)     dt  --  k  dz2, 

and  from  which,  having  once  obtained  the  coefficient  k2,  the 
true  radiation  -~,  loss  of  heat  in  the  unit  time,  may  be  computed. 

For  our  present  purpose  we  need  only  admit  that  the  rate  of 
radiation  is  proportional  to  the  second  differences  A2  (Qi  —  QQ) 
per  thousand  meters,  to  be  able  to  make  important  inferences. 
The  two  cases  of  low  I\  and  high  72  temperatures  in  the 
isothermal  region  are  considered  in  connection  with  average 
conditions  in  the  convectional  region  C;  between  the  isothermal 
and  the  convectional  region  there  is  a  region  of  absorption  and 
reflection  of  the  radiation  R,  wedge-shaped,  about  6,000  meters 
deep  over  the  equator,  and  thin  or  vanishing  in  the  arctic  zone; 
below  the  convectional  region  there  is  a  second  stratum  of 
absorption  and  reflection,  less  than  2,000  meters  deep,  and 
occupying  the  region  of  the  specific  diurnal  convection  R2.  We 
are  not  now  concerned  with  the  absorption  regions  RI,  R%,  but 


VALUES  IN  THE  RADIATION  EQUATION  291 

only  with  the  radiation  regions  /i,  72,  C.  Taking  the  mean 
values  of  A2  (Qi  —  Q2)  in  these  regions,  respectively,  and  the 
ratios  I\/C  and  /2/C,  together  with  the  means,  it  results  that 
about  2.11  times  as  much  radiation  is  passing  through  the  iso- 
thermal region  as  passes  through  the  convectional  region.  This 
can  only  signify  that  of  the  incoming  radiation  52  per  cent  is 
reflected  before  reaching  the  surface  of  the  earth,  and  that  a 
nearly  equal  amount  returns  to  space  as  albedo.  If  we 
again  admit  that  the  1.918  calories  determined  as  the  max- 
imum passes  below  the  cirrus  levels  to  the  lower  stations,  it 
follows  that, 

2.11  X  1.918  =  4.047  calories, 

is  the  approximate  average  value  of  the  "solar-constant."  This 
confirmation  of  the  results  from  the  bolometer  ordinates  also 
strengthens  the  conclusion  that  the  solar  temperature  of  emission 
is  about  6,900°.  Since  I0.r  =  1.918  calories,  and  this  corresponds 
nearly  with  the  temperature  5,800°  in  Table  65,  the  sum  of  the 
ordinates  for  a  full  radiator  at  5,800°,  summing  them  from 
/I  =  0.20 //  to  A  =  2.50 //,  is  about  27.28.  Now,  this  sum  is 
proportional  to  the  area  of  the  energy  spectrum  between  these 
wave  lengths,  beyond  which  limits  the  energy  not  included  is 
very  small  in  amount,  and  since  we  require  to  employ  2.11  times 
this  energy, 

2.11  X  27.28  =  57.56, 

it  is  seen  that  the  sum  of  the  ordinates  equal  to  this  amount 
lies  between  6,800°  and  6,900°  solar  temperature.  If  an  allow- 
ance be  made  for  selective  absorptions  in  the  several  zones  of 
the  earth's  atmosphere,  as  in  the  cirrus  region,  it  is  evident  that 
these  general  conclusions  may  be  extended  to  many  important 
special  researches. 

We  have,  therefore,  besides  the  negative  argument  that  the 
Bouguer  formula  is  indeterminate,  so  far  as  fixing  the  level  from 
which  the  observed  radiation  proceeds  to  the  surface,  the  three 
positive  arguments  that  the  solar  constant  is  4.00  calories  up- 
ward, and  the  temperature  of  the  solar  photosphere,  6,900° 


292          RADIATION,   IONIZATION,   AND  MAGNETIC  VECTORS 

upward,  namely,  (l)  The  maximum  1.918  calories  on  the 
10,000-meter  plane,  from  stations  at  different  elevations  above 
the  sea  level.  (2)  The  form  of  the  bolometer  ordinate  curves 
of  the  energy  spectrum  for  6,900°  solar  temperature.  (3)  The 
thermodynamic  ratio  2.11  of  the  radiation  in  the  isothermal  and 
convectional  regions.  Compare  in  addition  Chapter  VII. 

The  Measures  of  the  lonization  of  the  Atmosphere 

There  are  two  types  of  instruments  for  measuring  the  ioniza- 
tion  of  the  atmosphere,  (1)  the  Elster  and  Geitel  apparatus  for 
the  coefficient  of  dissipation  of  electric  charges  from  a  body 
connected  with  an  electroscope,  (2)  the  Ebert  ion-counter  with 
fixed  capacity  and  variable  charge,  and  the  Gerdien  ion-counter 
with  fixed  charge  and  variable  capacity.  The  former  measures 
the  coefficient  of  dissipation  A,  and  the  two  latter  measure  the 
component  parts  of  /,  namely,  the  number  of  ions  per  cubic 
centimeter  of  air  n,  and  the  velocity  of  motion  of  the  ions  which 
dissipate  the  charge  on  the  body  attached  to  the  electroscope. 
For  each  kind  of  electricity,  we  have, 

A  =  (e  n±  u+ 


where  e  =  3.4  X  10~10,  the  constant  charge  on  the  single  atom. 
These  subjects  have  been  discussed  in  many  papers,  and  the 
results  are  collected  in:  Die  Atmospharische  Electrizitat,  H. 
Mache  und  E.  V.  Schweidler,  Braunschweig,  1909;  Die  Luftelek- 
trizitat,  Albert  Gockel,  Leipzig,  1908;  Electricite  Atmospherique, 
Observatoire  de  1'Ebre,  1910,  par  le  P.  /.  Garcia  Molld,  S.J. 

There  is  a  fundamental  discrepancy,  amounting  to  about 
300  per  cent,  between  the  resulting  values  of  A  observed  and 
computed  by  the  two  methods  indicated,  and  as  it  is  our  purpose 
to  come  to  this  problem,  we  shall  as  briefly  as  possible  summarize 
the  formulas  and  the  conditions  of  the  instruments  leading  to 
these  conclusions.  The  elementary  electrostatic  formulas  can 
be  verified  by  reference  to  numerous  treatises,  of  which  one  of 
the  most  convenient  is,  Elements  of  the  Mathematical  Theory  of 
Electricity  and  Magnetism,  by  /.  /.  Thomson,  Cambridge,  1895. 


IONIZATION  OF  THE  ATMOSPHERE  293 

It  is  not  necessary  further  to  explain  the  individual  formulas 
which  follow  by  simple  transformations  from  the  elementary 
conditions  that  have  been  stated. 

Notation  and  Elementary  Relations 

The  sign  -f  signifies  repulsion  along  the  line  joining  two 
charges  of  the  same  kind  Q  Qf. 

Q  =  charge  on  a  point,  Qi  =  charge  on  unit  line,  Q2  =  charge 
on  unit  area. 

00' 

(693)  Force  =  F  =  ^^-.      Fn  =  the    normal,    and    Ft  =  the 

tangential  components. 

(694)  Total   intensity  =  I  =  S  FnS  =  /  Fn  d  S  =  4  IT  Q  on 

the  closed  surface  S. 

The  induction  is  zero  when  the  charge  is  outside  S. 
Special  applications  to  given  surfaces  for  unit  lengths. 

Sphere  Cylinder 

TotalforceF=Fg.47rr2  =  IwQ          F  =  Fc2irr  =  lirQi 

(695)  F.=  -£  Fc- 

Infinite  Plane 


Fp  =  47r<22 

The  difference  of  potential  A  V  =  the  work  done  in  moving 
the  unit  charge  against  the  electric  field  of  intensity  Fn  through 
the  distant  A  r. 

(696)     Work.  W  =  f*Fn  dr  =  Fn(r2-  r,)  =  (Vi  -  F2) 

for  the  unit  charge. 


for  the  charge  Q. 


294          RADIATION,   IONIZATION,   AND  MAGNETIC  VECTORS 

(697)  Force.  Fn=  -  (F2~Fl)  =  -  ^.     Mechanical 

f  2  —  f\  a^ 

force  F  =  ±FnQ. 

(698)  Work.  W  =  /Fndr  =  --  f^dr  =    y  = 

(Fi  -  F2)  for  the  sphere. 

(699)  Inner  energy.   U  =  J  JF  =  J  Q  (7i  -  F2)   = 


The  following  quantities,  Q,  Z>,  77,  <r,  w,  are  synonyms  for 
the  same  physical  process  in  a  dielectric,  and  differ  only  by 
coefficients. 

E  =  the  electric  force,  c  =  the  specific  conductive  capacity, 

H  =  the  magnetic  force,  /*  =  the  magnetic  permeability, 

D  =  cE  =  the  electric  displacement;  B  =  /J-H  =  the  magnetic 

induction.     77  =  -  —  Fn  =  the  polarization  =  <s  the  surface  den- 
sity =  Q  the  charge. 

(670)     U  =  ±W  =  ±D  E  =  \cE^=\^=\E(V,  -  F2). 

L> 

In  a  medium  other  than  air  the  specific  inductive  capacity 
is  K.  The  number  of  Faraday  lines  =  n. 


(671) 


— 


K  K 

(672)  Ui  =  |  .  —  Fn2  =  —  .  n  Fn  =  the  longitudinal  tension. 

K  K 

(673)  Ut  —  %  .  j—  .  F^2  =  —  .  w  ^  =  the  transversal  tension. 

A  very  complete  collection  of  electrical  and  magnetic  for- 
mulas may  be  found  in  Bulletin  I,  U.  S.  W.  B.,  1902,  "Fclipse 
Meteorology  and  Allied  Problems." 


ELECTROSTATIC  RELATIONS  PER  UNIT  LENGTH 

Electrostatic  Relations  per  Unit  Length 
Q  =  c  V  =  <*S 


295 


Formula 

Sphere 

Cylinder 

Plane 

(674)  Charge      Q  =  C  V 

0  =rF 

Ol=2lol~r 

»-if, 

(675)  Potential  V    =  Q/C 

(676)  Capacity  C  =  Q/V 

(677)  Potential 
coefficient  £  =  V/Q 

dV 
(678)  Force         F  =  -  — 
a  r 

(679)  Work        U  =  i  0  F 

C   =r 

1 

*  =  7 
*-I| 

F  =  2  Oi  loge  r 

F    =4*  r.  Qj 

~21oger 
£    =  2  loge  r 

I/  =  Oi2  loge  r 

I/  =  2  *  r  Q,* 

I/=i>Q'=*CF' 

Formula 

Two  Concentric 
Spheres 

Two  Coaxial 
Cylinders 

Two  Parallel 
Planes 

(680)  Potential  Fi  -  Fj 

Vl  _  Vt  =  Q  !ii  n 

Vi  -  Fz  = 

F!  -  Fz  = 

2Q,  i0g    11 

4  TT  Qz  (ri  —  rz) 

Q 

nr* 

1 

1 

(681)  Capacity  C  -  Fl  _  Fj 

C  —  ' 
rt-n 

21ojr    — 

4  TT  (ri  -  rz) 

(682)  Charge     Q  =  C  (Fi  -  Fz) 

Q=^(VI_VI} 

1 

1 

Ol  "              ri 

4  TT  (n  -  rz) 

2  loge  — 

(Fi  -  Fz) 

(Fi  -  Fz) 

(683)  Potential              Fi  -  Fz 

ri  -  r\ 

P  =  4  TT  (ri  -  rz) 

coefficient                Q 

nrj 

p  -  2  loge  rj 

(684)  Work  U  =  $  Q  (Vi  -Fz) 

n  rz 

U  =  log,,  —  Q.2 

U  -  2  rr  (n  _rj)  Qj* 

U  =  i  P  Q2  =  i  C  (Fi  -^Fz)2 

Q 

Oi 

02 

(685)  Surface  density  <r  =  — 

2  IT  (ri  -  rz)  / 

~   5 

(686)  Force    F  =  4  TT  o- 

r         Q 

F  =  (n^i 

4^Qz 

(rt-r-P) 

"S 

Quantity  neutralized   in  the 

unit  time 

(687)     j   idS  =4?rAQ  =oQ 

fidS-i*(e*u)Q 

fidS  =  Mnu)Ql 

JW5  -4-  («.«)& 

=  4  *  [(«««)  -t 

+  («nu)_jQ 

=  4irA  .  C   F    - 

Derivation  of  Coulomb's  Law  of  Dissipation 
(688)     fidS  =  4  TT  A  .  Q  =  4  w  A  .  ff  S  =  A.  F.  5  =  a  (X 


296          RADIATION,   IONIZATION,   AND  MAGNETIC  VECTORS 

(689)  fidS  =  -  -jj  .    The  loss  of  charge  with  the  time  dt. 

(690)  4  *  *  .  Q  =  -  ^2  .    Combining  (689)  and  (690)  in 

Coulomb's  Law, 

(691)  ±nx.dt= 


(692)  Vi  =  VQe~a(tl~^\ 

(693)  a  (/i  -  t0)  =  C.  log.  nat  -~  =  ^  .  C  log  ^-° 

Conduction  lonization  Currents 
Number  of  ions  per  cu.  cm.        n  +    n__ 
*  Charge  IP  E.  S.  U.  e+     e,     e  =  3.4  X  10  ~10. 

Velocity  of  ions  in  cm/sec  at  -          — ,  u  +     u  _ 

(694)  Conductivity  =  the  positive  current  through  1  cm2  in 
unit  time,  A  =  (e+  n  +  u+)  +  (e^n_  w_). 

Example  of  an  Insulated  Charged  Sphere  of  Radius  r 

(695)  Charge.  Q  =  CV  =  rV.  =  <?S. 

(696)  Surface  density,     a  =  - — -  =  -^.    S  =  the    surface    of 

the  sphere. 

(697)  Potential.  V  =  —  =  4  TT  r.  <r. 

(698)  Capacity.  C  =    r    =  ®. 

(699)  Force.  Fn  =  -^  =  4  TT  a,  acting    on    the   unit 

quantity  at  distance  r. 

*  Result  of  many  experiments.     Compare  Table  95. 


THE   ELSTER  AND    GEITEL   APPARATUS  297 

(700)  Ion  current  i  =  A  F  =  [(e  n  u)  +  +  (e  n  w)_]  — 

from  the  unit  area. 

(701)  On  whole  sphere        =  4  w  r2.  A  Fn  =  4  TT  [(e  n  u)  +  + 


This  ionization  current  neutralizes  the  charge  of  the  body  at 
a  certain  rate  by  4  w  (e  n  u)  +  Q  acting  outward,  and  4  x  (e  n  u)  _ 
Q  acting  inward,  as  to  the  surface. 

The  dissipation  coefficient  A  is  the  measure  of  the  rate  of  the 
neutralizing  the  charge. 

(702)  A+  =  4  TT  (neu)_  neutralized  negative  charge. 
/_  =  4  TT  (n  eu)+  neutralized  positive  charge. 

(703)  Quantity  neutralized  =  XQ  =  Xr  V  =  47r  Xr*  <r. 

Large  bodies  neutralize  faster  than  small  bodies. 
The  neutralization  amount  is  proportional  to  a. 

The  Coefficient  of  Electrical  Dissipation  of  the  Atmosphere  X 

The  Elster  and  Geitel  apparatus  consists  of  an  electroscope, 
whose  capacity  by  itself  is  C,  and  a  small  dissipating  body, 
generally  a  copper  cylinder  10  cms.  long  and  5  cms.  in  diameter, 
whose  capacity  by  itself  is  K.  The  dissipating  body  charged 
with  the  quantity  of  electricity  Q,  in  the  time  d  t,  loses  the  quan- 
tity —  d  Q,  by  the  formula  for  voltage, 


(704)  --(C  +  *). 

Since  it  is  required  to  know  the  rate  of  dissipation  from  the 
cylinder  alone,  where, 

(705)  Q  =  KV, 

we  construct  the  Coulomb  law,  by  division, 

,7r¥^  1    dQ  C  +  K  dV 

(706)  4,*  =  «=---      -=        --    -. 


298          RADIATION,   IONIZATION,  AND  MAGNETIC  VECTORS 

Multiply  by  d  i  and  integrate  for  common  logarithms, 

(707)  0  «,-<„)=  ^J^  "^  '  log  f°. 

This  is  the  formula  for  dissipation  from  the  body  to  the 
free  air,  provided  there  is  no  leakage  into  the  interior  of  the 
electroscope.  This  is  expressed,  from  parallel  reasoning,  for 
C  and  K  separately, 

(708)  <*(/!-/<>)  =-§^log^7- 
Combining  these  two  parts  into  one  formula, 

(709)  4^-.-^.£±*.£.(iog£;- 

C  F0' 


If  the  instrument  is  charged  with  the  electroscope  alone  for 
Fo'  at  the  time  /0,  then  the  V\  is  to  be  read  at  the  time  fc,  gener- 
ally an  interval  of  15  minutes  for  (/i  —  /0)-  When  the  insulation 
is  good  this  correction  is  small,  and  the  compensation  can  be 
made  by  (Fi)  =  Vi  +  A  Fi.  It  is  customary  to  express  the 
constant  a  in  terms  of  percentage  of  loss  of  charge  per  minute,  so 
that, 

..       ,       100    K  +  C     1  Fo 

(710)    a       lg-  T'M'log(FO- 

Finally,  the  general  formula  for  A  becomes  in  E.  S.  U., 

-2- 


4  TT       4  TT  X  60  X  100       4  TT  X  60  X  100        K 

-  log          =  (e  n+  u++  e  n_  u,)  300. 


The  Ebert  ion  counter  consists  of  an  electroscope  to  which  a 
condenser  is  attached  as  an  integral  part,  and  through  this  a 
known  volume  of  air  A  in  cubic  centimeters  per  second  is  drawn 
by  a  revolving  turbine.  The  capacity  C  of  this  combination  is 
known.  In  the  axis  of  the  long  cylinder  of  radius  r,  a  small 


THE  EBERT  ION  COUNTER  299 

cylinder  of  radius  rz  and  length  /  is  supported  by  an  insulating 
rod  connecting  with  the  electroscope.  If  the  apparatus  runs 
for  a  given  time  (t\  —  tQ)  while  the  small  cylinder  is  without 
charge,  there  will  be  the  loss  in  voltage 

(712)  C  (Fo'  -  F/),  (without  charge). 

When  the  rod  is  charged  the  loss  in  the  same  interval  is, 

(713)  C  (Fo  -  FO,  (with  charge).      . 

This  change  in  the  potential  is  caused  by  the  passage  of  N 
ions,  each  with  charge  e,  in  the  volume  of  air  A  per  second,  so 
that  in  volts  with  the  factor  300,  we  have  the  equation,  con- 
taining correction  for  the  leakage  by  (695) ,  for  positive  ions, 

(714)  +  e  N+  A  300  =  C  [(F0  -  FO  -  (F0'  -  F/)]  in   cubic 

meters. 

(715)  n+=NX  l<r6  =  C  [(F.  -  F:)  -  (Vtf  -  V,')}  jj~^  in 

cu.  cms. 

The  passage  of  n+  positive  ions  per  cu.  cm.  has  neutralized 
a  certain  amount  of  the  charge  of  the  opposite  sign,  so  that  by 
charging  F0,  F0'  for  one  sign,  the  number  of  ions  of  the  opposite 
sign  is  determined. 

Example  of  the  Computation 

A  1  run  =  0.185283  cubic  meters.     C  =  9.69. 
Ab  (560  seconds)  =  0.926417  cubic    C  X  2941 

meters.       300,4 

(+)     (Fo  -  FO  =  9.0.  (Fo'  -  F/)  =  0.6. 

For  positive  charge,  w_  =  84  X  102.54  =  861. 

(-)     (F0  -  FO  =  10.2  (Fo'  -  F/)  =  0.5. 

For  negative  charge,  n+  =  9.7  X  102.54  =  995. 

The  Formulas  for  the  Velocity  of  the  Ions 

The  velocity  of  motion  of  the  positive  and  the  negative  ions 
respectively  depends  upon  a  special  proposition,  as  follows: 


300          RADIATION,   IONIZATION,   AND  MAGNETIC  VECTORS 

J  E* 

(716)  =--U'  Gdr=-Fudx. 


(717)      F=-==-       ^       by  (678)  and  (682). 

2  loge 


H — *~^fc— *« 


FIG.  56.  The  parabolic  motion  of  the  ions  in  a  condenser  of  co-axial  cylinders 
charged  to  V.  u  =  the  velocity  of  motion  in  cm/sec.  F  =  the  radial 
component  along  r.  G  =  the  axial  component  along  x. 

(718)  T~  =  —  "TT   '   ~ 1>  f°r  the  unit  charge.     Integrate 

ax  Lr  r\ 

r  loge  —     after  transposing, 

^2 

2  ^     70  _  Vi 

(719)  r2  =  —  -^r  .  — ; x  +  C.    C  =  r02  on  ion  entering 

loge  —       the  condenser  field  at  r0- 

* 


(720) 


The  maximum  value  of  x  is  at  a  distance  less  than  /  from  the 
origin,  where  the  ion  entering  at  rQ  falls  finally  upon  the  charged 
surface  of  the  inner  electrode. 

2  loge  —  [Malls  inside  11   Condition 

,        v       r  „        -  x  -    ,  x  ^2  ,  ,     that  all  the 

(721)       —fr—  -  (fi'-r^)— rrr-  1  =falls  on  end  /  ionsfallupon 

electrode. 


L  <  falls  outside  / 


ri 


(722)       °~     '=  (r02  -  r22)  i  ',  equation  of  condition. 


velocity  apparatus  consists  of  the  primary  condenser 
and  electroscope,  used  for  counting  the  number  of  ions,  and  in 


EXAMPLE   OF  VELOCITY  COMPUTATION  301 

addition  an  auxiliary  condenser  in  the  same  axial  line,  the  inner 
electrode  charged  to  a  few  volts  A  V,  and  the  outer  connected  to 
earth. 

Let  VQ  —  Vi  =  the  loss  in  voltage  with  no  charge  on  the 

auxiliary. 

Fo'  —  Vi   =  the  loss  when  the  auxiliary  is  charged  to 

about    A  V  =  30   volts.    As   some   ions   fall   on    the 

auxiliary  electrode,  it  follows  that  (F0-  FI)  >  (F0'-  F/) 

The  amount  of  electricity  drawn  down  upon  the  auxiliary 

electrode  by  its  charge  of  opposite  sign  is, 


(723) 


We  have,  also,  e  n  =  C  (F0  -  7i)  =  F°  ~  V\    Hence, 


(Vo  -  Fi)  -  (Fo'  -  FiQ 


Example  of  the  Velocity  Computation 
A          0.85065  X  106  _  cubic  centimeters 


/i  — 10  515  per  second 

1  1  1 


47T/.AF"    47TX12X27       4071* 
~  •  2  log  ^  =  2.3026  X  2  X  (log  1.50  -  log  0.25)  =  3.584. 


1652  X  3.584 
Constant  =  -  -  =  1.45. 


Q-CFo'-F/),  Fo  -  Vl  auxiliary 

T7  -,7 

Vo  —  V\  not  charged. 

Fo'  -  Vi'  auxiliary 
charged. 


302          RADIATION,   IONIZATION,   AND   MAGNETIC  VECTORS 

(190.7  -  177.1)  -  (184.9  -  177.6) 

' 


(-)    Charge.    ,+  -  1.45 


1.45         f         =  0.67  cm/sec. 

lo.O 


(192.8  -  179.0)  -  (183.9  -  176.1) 
(+)    Charge,    u.  =  1.45  *  -        192.s  _  \79.0  -  '-  = 

1.45  —  '—  0        =  0.64  cm/sec. 

lo.o 

The  Gerdien  apparatus  for  the  number  and  velocity  of  ions. 
This  consists  of  a  double  condenser,  fitted  with  a  turbine  for  a 
measured  quantity  of  air  to  be  drawn  through  the  tubes  in  the 
unit  time.  There  are  two  electrometers,  /  with  a  variable 
capacity  C",  //  with  a  constant  capacity  C.  The  outer  cylinder 
has  the  radius  r\  =  4.9  cm.,  and  the  length  /  =  65  cm.;  the 
auxiliary  with  variable  capacity  C'  has  the  radius  r%  =  0.5  cm., 
and  /  =  20.1  cm.;  the  principal  with  constant  capacity  has 
the  radius  r2  =  1.0  cm.,  and  /  =  35  cm.  C  =  16.7,  C  =  20.2, 
in  some  instruments. 

The  number  of  ions.  The  capacities  are  both  at  a  minimum, 
and  the  number  of  ions  falling  on  both  electrodes  neutralizes  a 
quantity  of  electricity  of  the  opposite  sign,  expressed  by  the 
equation  (725),  where  A  =  the  number  of  cubic  centimeters  of 
air  that  passes  in  the  time  (/i  -  J0)  =  10800000  cm.3  for  80 
revolutions. 

(725)  e  n  A  300  =  C  (F0'  -  F/)  +  C  (VQ  -  Fi). 

(726)  n  =  [(C  (Fo'  -  F')  X  C  (F0  -  Fi)]  0.908. 

The  velocity  of  the  ions  per  second.  The  capacity  C'  is 
increased  to  C/,  from  16.7  to  124,  by  raising  the  capacity 
cylinder  25  divisions.  From  (722), 

v  '      v  '  2  loS*  7" 

(727)  K^ZL  =  (,2  _  ,2)  __-  . 

The  quantity  of  electricity  on  w-ions  entering  the  field 
through  the  area  TT  (r  02  —  rf)  falls  on  the  electrode,  so  that, 


GERDIEN  APPARATUS  303 

F' 


(728)     TT  (>o2  -  rf)  =  4  TT  /.  u.  C  V  =  4  w  I.  u. 


2  loge 


7-2 


Hence,  by  substitutions,  and  for  the  interval  d  t, 

V 


(729)     en  .TT  (r02  -  rf)  G d  t  =  e  n  .  4  TT  A  w 


2  **. 


C/       7     T// 
i  a  V  . 


(731)     u+  = 


en  + 


u_  = 


(e  n  u)  - 


Velocity  in  a 


(732)     A+  =  (en+  w_j_)  300. 


en- 
field  1  volt/cm.  E.  S.  U. 

?w_  «_)  300.     A  =  A  , +A_. 


EXAMPLE  OF  THE  GERDIEN  COMPUTATION 

COMPARE  MOLLA,  TORTOSA  OBSERVATORY 
(  —  )  Negative  charge  for  the  positive  ions  neutralized. 


I.     Ci  =  16.7 

II.     C  =  20.2 

H       m       s 

Index 

L 

B 

S           I 

ri        log  yi 

L          R 

S 

V 

to       9     40     18 
ti     10       6     33 

26     15 
li-h            1575 

log  (ti-io)  3.19728 

logcn_j_3.7254 
log«_j_     0.0912 
log  300     2.4771 

60 
140 

80 
A 

I 
3 
2 

11.2      10.7      21.9      19£ 
9.3        9.8      19.1      174 

yi^yi         2 

log.  log  VVVi1 
[Constant] 

>.8      2.30060 
LO      2.24055 

5.8     0.06005 

8.77851 
8.23544 

12.0      15.6      27.6 
9.0      10.5      19.5 

Vo-Vi 

Ci  (V(CVl\ 

209.4 
145.5 

63.9 
1290.8 
430.9 

1721.7 
Factor             0.908 

n_j.                  156.3 
Iogn+        3.19396 
loge             0.53148 
log  (ti-to)     3.19728 

log.  Const,  log  VoVW        7  .  01395 

log  en  i    (<i-fo)                      6  .  92272 
log  «+                                0.09123 
tt+                                     1.234 

=  1.967   X  KM 

log\+     6.29379  X  + 

log.  en  _j_  (h-to)     6 

.  92272 

Read  the  scales  for  V0'  (auxiliary),  and  F0  (principal),  while 
C  is  at  minimum  value  16.7.  Raise  the  C  to  C/  =  124  by  25 
divisions.  Start  the  turbine  and  read  the  hour  and  index;  after 
an  interval  for  80  A  read  the  hour  and  index.  Lower  the  capa- 


304  RADIATION,    IONIZATION,    AND   MAGNETIC   VECTORS 

city  to  minimum  C'  —  16.7  and  read  V\  and  V.     The  formulas 
become  by  evaluation : 
1 


u  = 


e  n  '  ti  — 
A  =  e  n  u  300. 
4  TT  A  =  a. 


1  V r 

0.0172  log  T/7  (auxiliary). 


Example  Continued. 


(  +)     Positive  charge  for  the  negative  ions  neutralized. 


16.7 


II.     C  =  20.2 


H       m      s 

Index      1 

R 

S 

V 

logy 

L          R           i 

?           V 

to     10     11     30 
h            37       5 

65      10 
145        9 

.0      10.7      20.7 
.5        9.5      19.0 

188.4 
173.4 

2.27508 
2.23905 

12.5      15.8      28 
9.9      11.7      21 

Vr-Vi 

C(Vo—Vi) 
Cl(Vol—Vi 

Factor 
w_ 
log  u 
loge 
log  (fa-fo) 

.3      213.9 
.6      162.0 

25     35 
ti-to               1535 

log  '(h-to)  318611 

log  en     3.60299 
logtt      0.00300 
log  300  2.47712 

logv_     6.08311 

80                                Vo^-Vi 
A 
log.  log.  V^/Vi 
[Constant] 

log  Const,  log. 

log  en_  (ti-to) 
log  u_ 
u_ 

A_=A  1.211X10-4    A=A  + 

i     15.0 

0.03603 

8.55666 
8.23544 

51.9 

1048.4 
0        250.5 

WVi1      6.79210 

6.78910 
0.00300 
1.007 

+  A_  =3.178X10-4 

1298.9 
0.908 
1179 
3.07151 
0.53148 
3.18611 

log  e  n_  (h-to)  6.78910 

The  Cause  of  the  Discrepancy  in  the  Values  of  the  Conductivity 
of  the  Atmosphere,  as  Determined  by  Two  Methods 

It  will  be  seen  from  the  collection  of  results  in  Table  68  that 
we  have  arrived  at  two  very  different  results  for  the  coefficient 
of  atmospheric  conductivity  X,  as  determined  by  the  two  methods 
that  have  been  described.  Section  I  is  compiled  in  part  from 
Mache  and  Schweidler's  volume,  and  in  part  from  Bigelow's  obser- 
vations of  1905,  made  during  the  U.  S.  Eclipse  Expedition  to 
Spain  and  Algeria,  and  of  1912  made  at  Cordoba,  Argentina, 
with  the  resulting  value  of /I  =  5.575  X  10  ~5  E.  S.  U.  Section 
II  is  compiled  from  the  same  sources,  with  the  result  A  =  20.49  X 
10  ~"5,  so  that  the  ion  counters  give  a  value  for  X  which  is  3.68  times 
greater  than  by  the  dissipation  apparatus.  The  cause  of  this 


DISCREPANCY   AS   DETERMINED   BY   TWO   METHODS 


305 


great  discrepancy  is  not  far  to  seek.  The  dissipation  apparatus 
is  generally  worked  under  a  cylindrical  hood,  which  converts  it 
into  a  condenser,  through  which  the  air  current  is  not  freely 
passing,  with  the  effect  that  the  free  dissipation  is  transformed 
into  a  saturated  or  stagnant  circulation  of  the  ions.  It  is  easy 
to  show  that  this  apparatus  with  the  hood  does  not  respond  to 
the  capacity  formula  upon  which  the  computations  are  based. 
We  have  used  cylinders  of  different  dimensions,  with  the  hood 

TABLE  68 

COMPARISON  OF  THE  COMPUTED  VALUES  OF  A  FROM  THE  DATA 
OBSERVED  BY  THE  Two  METHODS 


Section  I.     Elster  and  Geitel  dissipation  apparatus 


Stations 

al  + 

al_ 

2 

a  + 

0_ 

A  + 

A_ 

A 

Lugano     ... 

1.87 
4.22 
3.02 
3.76 
2.88 
1.17 
1.26 
0.84 
0.72 
0.97 
1.26 
0.55 
3.33 

1.36 
1.45 
3.63 
1.87 
0.73 
1.50 

2.59 

1.87 
4.44 
3.38 
5.86 
4.62 
1.54 
1.34 
1.33 
1.00 
1.29 
1.43 
0.60 
3.82 

1.39 
1.55 
3.77 
1.90 
0.79 
1.56 

2.53 

1.00 
1.05 
1.11 
1.55 
1.60 
1.32 
1.06 
1.58 
1.43 
1.36 
1.13 
1.09 
1.15 

1.02 
1.07 
1.04 
1.02 
1.08 
1.04 

0.98 

3.12 
7.04 
5.03 
6.27 
4.80 
1.95 
2.10 
1.40 
1.20 
1.62 
2.10 
0.92 
5.55 

2.27 
2.42 
6.05 
3.12 
1.22 
2.50 

4.32 

3.12X10"4 
7.40 
5.63 
9.77 
7.70 
2.57 
2.23 
2.22 
1.67 
2.15 
2.38 
1.00 
6.37 

2.32 
2.58 
6.28 
3.17 
1.32 
2.60 

4.22 

2.48 
5.59 
3.99 
4.98 
3.81 
1.55 
1.67 
1.11 
0.95 
1.29 
1.67 
0.73 
4.40 

1.80 
1.92 
4.80 
2.48 
0.97 
1.99 

3.43 

2.48 
5.87 
4.47 
7.75 
6.11 
2.04 
1.77 
1.76 
1.33 
1.71 
1.89 
0.79 
5.06 

1.84 
2.05 
4.98 
2.52 
1.05 
2.06 

3.35 

4.96  XlO-fi 
11.46 
8.46 
12.73 
9.92 
3.59 
3.44 
2.87 
1.28 
3.00 
3.56 
1.52 
9.46 

3.64 
3.97 
9.78 
5.00 
2.02 
4.05 

6.78 

Capri  

Tromso 

Spitzbergen,  land.  .  . 
Spitzbergen,  sea  .... 
Juist 

Wolfenbuttel  
Misdroy  

Ostsee 

Potsdam  
Kremsmiinster  
Triest  .... 

Karasjok  

Daroca,  1905  
Porta  Cceli  
Guelma.  . 

Bona  

Casar,  bulkhead.  .  .  . 
Ccesar,  hatch  

Cordoba  1912. 

5.575XKT8 

and  without  the  hood,  leading  to  the  following  results: 
cylinders  are  as  indicated, 


The 


306          RADIATION,   IONIZATION,   AND   MAGNETIC   VECTORS 

TABLE  68 — (Continued) 

Section  II.     Ebert's  and  Gerdien's  Ion  counters.     A  =  (en+u+  +  en_u_) 

300  E.  S.  U. 


Stations 

n  + 

n  _ 

u  + 

u_ 

*  + 

A_ 

A 

.          dv 
*  =  *dh 

Helgoland,  dune.  .  .  . 
Helgoland,  oberland. 
Swinemiinde       .... 

382 
735 
823 
1088 
1147 
1323 
1029 
1264 
1558 
1793 
559 
1176 
828 
1117 
1000 
588 

206 
382 
647 
882 
794 
1117 
853 
382 
1238 
823 
529 
1205 
676 
970 
735 
588 

Potsdam  

Atlantic  Ocean 

Seewalchen  
Mattsee     

1.02 

1.25 

13.76 

14.24 

28.00X1Q-8 

9.33X10-7 

Santis 

Miinchen 

Bayern  Alpental  
Golf  von  Lyon 

IMallorca 

0.83 

0.90 

9.96 

11.06 

20.02 

6.67 

Barcelona.       .      .    . 

Karasjok   

Freiburg 

1.00 

1.11 

10.20 

8.32 

18.52 

6.17 

Stiller  Ocean 

Gottingen  

1  32 

1  40 

Daroca  1905 

999 
2940 
1155 
621 
1013 
1275 

1558 
1733 

819 
2960 
972 
626 
1166 
1206 

1542 
1522 

0.684 

0.771 

6.97 

6.43 

13.40 

4.47 

Guelma. 

Bona  

CcBsar,  bulkhead.  .  .  . 
CcKsar  hatch 

Casar,  outward  
Cordoba,  Sep.  12  to 
Oct.  10,  1912  
Oct.  11-26  

0.701 
0.715 

0.613 
0.766 

10.77 
11.83 

10.30 
10.08 

21.07 
21.91 

7.02 
7.30 

20.49X1Q-8 

6.83X10"7 

The  value  of  A  by  the  Ebert  or  Gerdien  methods  is  more  than  three  times  greater 
than  by  the  Elster  and  Geitel  method. 

(1).     r  =  0.25cm.     (2).  r  =    2.50cm.     (3).  r  =    2.50cm. 
/  =  5.00  cm.  /  =  10.00  cm.  /  =  20.00  cm. 

K  =  0.65.  K0  =  5.46.  KQ  =  10.91. 

The  capacity  of  the  electroscope  is  4.36  cm. 
The  capacity  coefficients  without  hood  become, 
K+C  K  +  C  K  + 


K 


7.71. 


K 


1.80. 


K 


1.40. 


DISCREPANCY   AS   DETERMINED   BY   TWO   METHODS  307 

C  C 


K  +  C 


0-87. 


-0.44. 


=  0.29. 


The  capacity  coefficients  with  the  hood  become: 

K       =  1.34.  K       =  7.50.  K       =15.00. 


K  +  C 
K 

C 


=  4.25. 
=  0.76. 


K  +  C 
K 

C 
K  +  C 


=  1.58. 


0.37. 


K  +  C 
K 

C 
K  +  C 


=  1.29. 


=  0.23. 


Two  sets  of  experiments  have  been  carried  out,  one  in  1905 
and  the  other  in  1912.  In  the  Eclipse  expedition  the  hood  was 
used  throughout  the  observations,  but  the  dissipation  bodies 
(1)  and  (2)  were,  used  in  frequent  interchanges.  The  following 
table  shows  the  average  fall  in  scale  divisions  during  fifteen- 
minute  intervals,  with  (2)  the  cylinder,  and  (l)  the  small  rod, 
respectively: 

TABLE  69 
MEAN  Loss  IN  SCALE  DIVISIONS  IN  FIFTEEN-MINUTE  INTERVALS 


Station 

Daroca 

Porta  Coeli 

U.  S.  S.  Ctesar 
Bulkhead 

U.  S.  S.  Casw 
Hatch 

Cylinder  10x5  cm. 
Rod  3x0.5  cm. 

No.     (+)     (-) 
657    -5.4    -5.4 
139    -5.3    -5.5 

No.     +     (-) 
354    -5.8    -6.3 
80    -5.1    -5.8 

No.     (+)     (-) 
147    -3.4    -3.6 
32    -3.3    -3.4 

No.     (+)     (-) 
126    -6.5    -6.7 
25    -6.6   -6.6 

The  observations  at  Daroca  and  Porta  Cceli  were  continued 
throughout  the  twenty-four  hours.  It  is  seen  that  the  size  of  the 
dissipating  body  under  the  hood  does  not  effectively  control  the 
rate  of  dissipation,  and  that  the  computed  capacity  coefficient 
does  not  properly  enter  into  the  computation. 

A  series  of  experiments  was  made  at  Cordoba,  1912,  with 
the  three  dissipation  bodies,  in  part  with,  and  in  part  without, 
the  hood.  The  mean  values  of  the  loss  in  scale  divisions  are 
proportional  to  the  following  data: 


308          RADIATION,   IONIZATION,   AND   MAGNETIC  VECTORS 


TABLE  70 
MEAN  LOSSES  IN  SCALE  DIVISIONS  WITHOUT  AND  WITH  HOOD 


Dissipation  Body 

Without  Hood 

With  Hood 

(+) 

(-) 

Mean 

(+) 

(-) 

Mean 

(1)  Rod 

1  58 

1.32 

1.45 

2.07 

1.73 

1.90 

(2)  Small  cylinder  .  .  . 

4.82 

3.98 

4.40 

1.63 

2.07 

1.85 

(3)  Large  cylinder  .  .  . 

5.60 

5.12 

5.36 

2.20 

1.97 

2.09 

1.98 

1.92 

1.95 

It  is  seen  that  with  the  hood  the  size  of  the  dissipation  body 
is  an  indifferent  matter,  but  that  without  the  hood  the  size  of 
the  body  and  the  rate  of  dissipation  progress  together. 

A  special  observing  shelter  at  Cordoba  was  constructed  of 
canvas  and  netting,  so  that  the  dissipation  observations  could 
be  made  without  using  the  hood,  and  the  results  for  the  series 

are  recorded  in  Table  68.     Referring  to  the  capacity  coefficients 

/("   j   C* 

for  the  three  bodies,  it  is  seen  that  — •« —  increases  as  K  dimin- 
ishes, the  upper  limit  for  K  =  0  being  <*> ,  and  for  K  =  <*D  being 
1.  We  may,  therefore,  increase  the  coefficient  by  decreasing 
the  size  of  the  dissipation  body.  Since  most  of  the  experiments 


of  Table  68  were  made  with  the  hood  and  for 


K 


=  1.58 


(on  the  average),  while,  without  the  hood  in  Cordoba, 
1.80,  it  is  apparent  that  the  comparatively  large  value  A 


K  +  C 
K 

6.78  X 

1CT0  is  due  to  this  fact,  at  least  in  part.  We  can  make  a  cylinder 
for  a  proper  dissipation  body  which  will  give  5.575  X  3.68  = 
20.49.  Its  approximate  size  can  be  obtained  by  computation, 
but  its  actual  and  correct  size  is  a  matter  for  experiment.  Such 
experiments  were  executed  at  La  Quiaca,  1913,  with  the  result 
that  a  dissipating  cylinder  52  centimeters  long  by  1  centimeter  in 
diameter  will  give  nearly  the  same  value  of  A  as  the  ion-counters. 
Similar  results  have  been  obtained  in  Potsdam.  Such  a  long 
cylinder  is  very  difficult  to  charge  to  a  sufficiently  high  potential 
V  and  it  is  impractical.  The  Elster  and  Geitel  apparatus  with- 


DISCREPANCY  AS   DETERMINED    BY   TWO   METHODS  309 

out  hood  is  valuable  for  giving  relative  values  of  (a+  a_)  (A+  A_), 
and  not  their  absolute  values.  These  must  be  obtained  by 
means  of  the  Ebert  or  Gerdien  ion-counters. 

In  Terrestrial  Magnetism,  Vol.  XVIII,  No.  4,  Vol.  XIX,  No.  1, 
No.  2,  No.  3,  1914,  Mr.  W.  F.  G.  Swann  discusses  the  theories  of 
the  several  ionization  apparatus,  and  indicates  that  several  cor- 
rections must  be  applied  in  order  to  remove  certain  errors. 
These  depend  upon  the  following  conditions: 

1.  The  variation  'of  the  atmospheric  potential  gradient  with 
the  height  above  the  surface  to  the  axis  of  the  horizontal  cylinder. 
The  change  may  amount  to  20%. 

2.  The  change  in  the  shape  of  the  electric  stream  lines  on  en- 
tering the  cylinder,  due  the  electrostatic  attractions  and  repul- 
sions, whereby  the  number  of  ions  entering  may  vary  as  much 
as  20%  to  30%. 

3.  The  modification  of  the  capacity  of  the  inner  condenser, 
which  should  contain  the  stub  of  the  supporting  rod  outside  the 
electroscope  as  well  as  the  small  inner  charged  cylinder.    This 
may  increase  the  "computed  capacity"  into  the  "measured 
capacity"  by  as  much  as  30%  to  40%. 

4.  The  value  of  the  unit  electric  charge,  as  determined  by 
several  different  lines  of  experiments,  has  been  taken  3.4  X  10~10, 
but  the  electrical  constant   of    Table  95,   e  =  4.774  X  10~10, 
seems  to  require  this  increase  in  the  computation  of  the  coefficient 
of  conductivity  ^.     It  is  evident  that  the  entire  subject  is  in 
need  of  further  discussion. 

The  Atmospheric  Electric  Potential' 

The  distribution  of  the  electricity  in  the  atmosphere  consists 
normally  of  a  heavy  positive  charge  of  several  thousand  volts  at 
about  5,000  meters  altitude,  which  diminishes  toward  the 
earth,  with  an  increasing  potential  fall  near  the  surface,  the 
latter  being  charged  with  a  negative  potential.  There  are 
continual  fluctuations  among  these  potentials  in  diurnal,  annual, 
and  other  periods,  or  in  aperiodic  variations  as  in  thunder- 
storms and  minor  changes. 


310          RADIATION,   IONIZATION,   AND   MAGNETIC  VECTORS 


Let  V  =  the  potential  at  any  height. 

d  V 

(733)  Force  F  =  —  -rr  =  4  TT  <r,  where  <r  is  the  surface  den- 

sity.    (699) 

(734)  —  -T^T  =  +  -jjp  =  —  4  TT  p,  where  p  is  the  volume 

density. 

_&V__        volts  volts  E.  S.  U. 

d  h  ~     ~  meter  ~      "  100  cm.          "  300  X  100  cm.' 


Surface  density  for  the  average  potential  fall  of  100  volts /m. 

100 
4  TT  X  300  X  100 


1 00 
(735)     *  =  -  7  —  =  -  2.65  X  1<T4  E.  S.  U. 


d2  V 
The  potential  gradient  changes  at  about  —  1/1000  =  -T-VT. 

d   fl 

(736)  ^=-f|-41;=-1^56(l^)35o  =  +27X 

10~9  E.  S.  U. 

The  total  surface  charge  of  the  earth  is  computed  from  <r 
with  the  radius  of    the  earth  r  =  6.37  X  108  cms.,  by  (696), 

(737)  QE  =  4  TT  r2.  *  =  12.56  X  (6.37  X  108)2.  (-  2.65  X  10~4) 

=  -  1.35  X  1015. 

Since  1  coulomb  =  3  X  109  E.  S.  U., 
QE  =  -  4.5  x  105  E.  S.  U. 

7^2  1  c*      2     2 

The  electric  pressure  Ut  =  Ut  =  -  -  =  —  ^—  -  —  2  TT  <r2,  by 

O  7T  O  7T 

(672). 

(738)  2  TT  **  =  4.43  X  10~7  dynes. 

The  vertical  electric  current  for  A  =  20.5  X  10~5  E.  S.  U. 

- 

E.  S.  U.  by  (700). 
i  =  2.05  X  10~15  amperes/cm.2,  since  1  ampere  = 

3  X  10~9  E.  S.  U. 


(730)     ^.^xHT.-e.  83X10 


ATMOSPHERIC  ELECTRIC  POTENTIAL 


311 


The  theory  adopted  in  this  work  of  the  electric  potential  and 
its  gradient  observed  in  the  lower  atmosphere  is  that  the  in- 
coming radiation  ionizes  the  aqueous  vapor  in  the  strata  within 


Height 
Z 


5000 


4000 


3000 


2000 


1000 


000 


Gradient  per  meter = 


Voltage  at  different  heights 

1602  x  100       1236  x  100 


0 60 


Volts  per  meter  =  -r-^- 


Volts 


FIG.  57.     The  electric  potential  fall  and  the  voltage  at  different  heights. 

a  few  thousand  meters  of  the  surface,  so  that  the  normal  charge 
is  150,000  volts  at  5,000  meters,  100,000  volts  at  1,400  meters, 
and  0  volts  at  the  surface,  with  negative  induction  in  the  earth 
itself.  A  study  of  the  radiation  data  of  Table  67,  those  of  Table 
63  for  the  distribution  of  the  aqueous  vapor,  together  with  those 
of  Fig.  57  for  the  electric  potential,  may  lead  to  the  function 
connecting  radiation,  ionization,  aqueous  vapor  contents,  electric 
currents,  and  diurnal  magnetic  deflecting  vectors.  This  subject 
will  require  prolonged  research  in  observation  and  analysis. 
Table  57  gives  the  distribution  in  heights  for  two  cases  of  the 
voltage  (160200,  123600),  and  the  voltage  gradient  per  100 
meters  (-8  -5)  at  5000  meters,  with  (-130  -90)  at  the 
surface  where  V  =  o  in  both  cases. 


CHAPTER  VI 
Terrestrial  and  Solar  Relations 

The  Five  Types  of  the  Diurnal  Convection  in  the  Earth's  Atmosphere 

THE  further  analysis  of  the  problems  of  electric  and  magnetic 
variations  depends  upon  the  determination  of  the  types  of  the 
diurnal  convections  in  the  earth's  atmosphere.     There  are  five 
of  these  types,  distinct  from  one  another:    (1)  In  the  Arctic 
zone;    (2)  in  the  North  Temperate  zone,  Lat.  +  66°  to  +  30°; 
(3)  in  the  Tropic  zone,  Lat.  +  30°  to  -  30°;    (4)  in  the  South 
Temperate  zone,  Lat.  -  30°  to  -  66°,  and  (5)  in  the  Antarctic 
zone.     What  is   needed   is   a   complete   determination   of   the 
diurnal  deflecting  wind  vectors,  for  each  hour  of  the  day  and 
night,  and  on  several  planes  from  the  surface  to  3,000  meters,  as 
000,  200,  400,  600,  800,  1,000,  1,500,  2,000,  2,500,  3,000  meters. 
These  can  be  obtained  by  kites  or  captive  balloons,  but  the  labor 
will  be  not  inconsiderable.     Unfortunately  the  available  material 
is  very  meager,  and  it  is  almost  wholly  lacking  during  the  night. 
Without  the  night  observations  those  made  during  the  day,  6 
A.M.  to  6  P.M.,  are  of  quite  subordinate  value.     At  present  we 
have  data  made  during  the  day  and  night  only  at  the  Blue  Hill 
Observatory,  1897-1902,  analyzed  in  my  papers,  "  Studies  on 
the  Diurnal  Periods  in  the  Lower  Strata  of  the  Atmosphere," 
Monthly  Weather  Review,  February  to  August,  1905;  at  several 
Mountain   Observatories,   J.  Hann,  K.  Ak.  Wiss.,  Wien,   Bd. 
CXI,  Abth.  Ha,  December,  1902,  and  April,  1903;    there  are 
several  stations  on  the  surface  which  can  be  utilized  for  pro- 
visional discussions,  Wien,  Mauritius,  Batavia,  Cordoba,  Cha- 
carita,  Laurie  Island  in  the  South  Orkneys.     The  circulation  in 
the  North  Temperate  zone  can  be  quite  accurately  constructed, 
while  that  in  each  of  the  other  zones  can  only  be  provisionally 
inferred.     The  available  data  have  been  thoroughly  recomputed 
with  the  results  collected  in  Table  71,  where  s  =  the  velocity 
in  meters  per  second,  and  &  =  the  azimuth  angle  from  5  =  0° 

312 


FIVE   TYPES    OF   DIURNAL   CONVECTION 


313 


| 

M 

E 

H 
H 
B 

I 


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3  o 
O  2 

<N 


I  § 

O       CO 

c/5 


E 


L 


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314 


TERRESTRIAL   AND    SOLAR  RELATIONS 


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i—  1  0}   CO  00  CO  CO  r-(i-( 


rHTtlOOOOlNC'O-tf 
rH    CO    ^^    ^O    ^O    CO    C^    ^^ 

CO          ^™H  ^H  *"^  C^  CO  CO 


OOOOOOOi-HrHOOO 


OOOOO^O 
iOcOCDcOrJH 


COlMOO^Oi 


OOOOOO'-H'-HOOOO 


^ 
<M 


(N  <M  <N  <M  r-i 


oooooooooooo 


88  S  8  S  848  g  2  3  fc  8 

CO  (M  (N  C^  (N  -r-i 


oooooooooooo 


FIVE   TYPES    OF   DIURNAL   CONVECTION 


315 


I   a 

i\ 

S  i 

PQ     o 

S§ 

C/5 

i 

a 


8  s 

M  C 

S  s 

(/) 


2«* 

H         0 


^CO 
CO  <N 


000000000000 


T~^  O^  CO  CO  ^H  OO 

i—  (JO^fCOOOs 

COCOCMfNr-t 


0000 


^D 
(M 


*O  "^  CO  CO  ^"* 


co  co       1-1 


»Ot-HT-iC<ICO<N'*iOiOC5 

THi—  IdCOCOlOi—  IT—  (OOS 
THi-Hi—  li—  IT—  ICMCOCOCOC^ 


t>*  CO  O^  i^  *O  C^l  ^O  CO  C^  CO  O5  i^ 
(N<N<MCOCOCOCOCO(N'-<COCO 
^Hr-li-lT-lrH(NCOCOCOCO  i-l 


CO  CO  CO  CO  CO 


316 


TERRESTRIAL  AND    SOLAR  RELATIONS 


through  E  =  90°,  N  =  180°,  W  =  270°.  The  vector  direction 
is  that  toward  which  the  stream  is  moving.  Having  the  mean 
observed  hourly  vectors,  D,  and  the  mean  24-hourly  resultant 
R,  these  deflecting  vectors  D  are  such  that  R  +  D  =  0,  con- 
structed as  true  vectors  of  velocity  and  direction. 


MN. 


10      Noon      2 


10       MN.       2 


From  Stations  in  the  Northern  Hemisphere 


\ 


\\ 


\ 

\ 


\ 


\ 


\ 


\ 


\ 


\ 


\ 


From  Stations  in  the  Southern  Hemisphere 


FIGS.  58  and  59.    The  probable  types  of  the  mean  diurnal  circulation. 


The  data  of  Table  71  were  plotted  in  diagrams  and  the 
apparent  circulation  for  the  North  Temperate  zone  was  con- 
structed, as  in  Fig.  58.  By  analogy,  from  the  surface  data  of 
Cordoba  and  Chacarita,  the  circulation  is  constructed  for  the 


DIURNAL  VARIATIONS   OF  ELEMENTS  317 

South  Temperate  zone,  Fig.  59.  From  Mauritius  and  Batavia, 
we  have  that  from  the  Tropics;  and  those  for  the  Arctic  and 
the  Antarctic  zones  are  made  from  other  data  to  be  mentioned 
later.  In  the  two  Temperate  zones  the  circulation  is  oppositely 
directed,  in  general,  with  turning-points  at  10  A.M.  and  8  P.M., 
with  divergence  at  2  to  4  P.M.,  and  convergence  at  6  to  8  A.M., 
and  8  to  12  P.M.  The  air  is  rising  in  the  afternoon  and  falling 
in  two  streams  during  the  night,  as  10  to  12  P.M.  and  2  to  8  A.M. 
This  is  the  circulation  which  is  caused  by  the  diurnal  heating  of 
the  earth's  surface  and  the  lower  strata  of  the  atmosphere  with 
rising  air  during  the  daytime,  and  with  cooling  and  descending 
air  during  the  night.  This  circulation  is  limited  to  3,000  meters 
from  the  surface,  and  it  is  not  vigorous  above  2,000  meters. 
The  local  conditions  of  mountain  stations  introduce  many 
minor  modifications,  and  such  stations  are  never  fully  equivalent 
to  ideal  free-air  conditions.  The  diagrams  of  Figs.  58,  59, 
contain  the  horizontal  component  chiefly,  but  the  vertical 
component  can  be  approximately  inferred  from  the  general 
stream  lines.  Similarly,  we  have  the  horizontal  and  the  vertical 
circulations  in  the  several  zones,  as  may  be  seen  in  Fig.  64  in 
connection  with  the  magnetic  vector  systems  which  depend 
upon  them.  The  vectors  of  Fig.  64  are  the  ends  of  the  stream 
lines  at  the  surface,  as  determined  by  the  data  of  Table  71. 
These  data  were  actually  applied  to  globe  models,  and  from 
them  the  adopted  circulation  was  derived. 

The  Diurnal  Variations  of  the  Meteorological,  Electrical,  and 
Magnetic  Elements 

The  effect  of  the  diurnal  circulation  on  the  several  meteor- 
ological elements  is  very  complex,  and  especially  so  in  view  of 
the  incessant  interchanges  between  the  diurnal  and  the  semi- 
diurnal periods  in  the  lower  strata.  Table  72  summarizes  some 
examples  of  this  interconversion,  which  can  be  profitably  studied 
by  transferring  the  data  to  suitable  diagrams. 

1.  The  temperature  data  for  the  lower  strata,  000  to  2,500 
meters,  are  taken  from  Table  24;  that  for  B  is  from  the  same 


318 


TERRESTRIAL  AND    SOLAR   RELATIONS 


table,  by  the  conversion  from  P  to  B.  The  temperature  has  a 
simple  diurnal  wave  at  the  surface,  as  heretofore  explained,  but 
a  semidiurnal  wave  above  500  meters,  diminishing  to  extinction 
at  about  3,000  meters.  These  results  conform  to  the  Blue  Hill 
direct  observations  of  temperature  in  the  free  air. 

TABLE  72 

EXAMPLES  OF  THE  TRANSITION  FROM  THE  SEMIDIURNAL  PERIODS  TO  THE 
DIURNAL  PERIOD  OF  DIFFERENT  ELEMENTS. 


1.  Temperature  T 

2.  Barometric 

Pressure  B 

•\  jf  _4._ 

Cordoba 

TVyT^4-« 

Cordoba 

Meters 

000 

500 

1000 

1500 

2500 

Meters 

000 

500' 

1000 

1500 

2500 

0  A.M. 

292.8 

295.0 

291.0 

287.7 

280.9 

0  A.M. 

760.35 

717.64 

676.95 

638.18 

565.99 

2 

291.7,295.4291.9 

288.6 

281.0 

2 

760.13717.27676.72 

638.10 

565.98 

4 

291.1295.3,291.6288.0 

281.0 

4 

760.40  717.42  676.85 

638.18 

565.98 

6 

291.0294.9290.6287.5 

280.8 

6 

760.80 

717.84 

677.16 

638.29 

565.99 

8 

293.  2|294.  0290.  0287.1 

280.8 

8 

760.93 

718.22  677.30 

638.33 

566.00 

10 

297.71293.5289.5287.0280.7 

10 

760.95 

718.28677.34 

638.34 

566.01 

0  P.M. 

300.2295.0290.7287.7,280.8 

OP.M. 

760.30 

717.80677.06 

638.28 

565.99 

2 

301.3298.0292.61288.8 

280.9 

2 

759.08 

717.05676.71 

638.13 

565.97 

4 

300.7 

298.6293.8289.7 

281.2 

4 

758.56 

716.72  676.43 

637.93 

565.95 

6 

299.0297.7 

293.5 

290.0 

281.5 

6 

758.62 

716.49  676.25 

637.79 

565.94 

8 

296.4296.1 

292.2 

288.9 

281.2 

8 

759.70717.15676.60 

638.02 

565.97 

10 

294.3 

294.7 

290.7 

287.1 

280.8 

10 

760.58 

717.91 

677.21 

638.27 

566.00 

3.  Vapor  Pressure  ed 


4.  Vapor  Pressure  ej 


Tower 

Sal  ton  Sea,  Tower  No.  1 

Tower 

Salton  Sea,  Tower  No.  4 

Pans 

(1) 

(2) 

(3) 

(4) 

(5) 

Pans 

(1) 

(2) 

(3) 

(4) 

(5) 

Feet 

00 

10 

20 

30 

40 

Feet 

2 

10 

20 

30 

40 

0  A.M. 

11.7 

11.7 

11.6 

11.8 

12.9 

0  A.M. 

16.0 

14.3 

14.8 

14.2 

14.3 

2 

10.7 

11.0 

11.0 

11.0 

12.7 

2 

14.8 

13.8 

14.1 

13.3 

13.5 

4 

10.0 

10.2 

10.3 

10.6 

12.5 

4 

13.7 

12.8 

13.2 

12.0 

12.7 

6 

9.8 

10.2 

10.1 

10.4 

12.3 

6 

12.6 

12.7 

12.9 

11.6 

12.5 

8 

12.4 

13.2 

12.2 

12.3 

13.3 

8 

14.6 

14.6 

14.0 

13.7 

14.1 

10 

14.3 

14.7 

13.6 

13.7 

13.9 

10 

16.9 

16.5 

15.8 

15.0 

15.4 

0  P.M. 

12.1 

12.1 

10.6 

11.2 

12.0 

0  P.M. 

18.0 

17.7 

16.1 

13.8 

14.0 

2 

9.7 

9.6 

8.2 

8.5 

9.5 

2 

18.6 

17.3 

15.5 

12.0 

12.0 

4 

9.3 

9.3 

8.4 

8.2 

8.9 

4 

18.8 

16.4 

14.6 

11.8 

11.3 

6 

9.9 

9.7 

9.8 

8.7 

8.6 

6 

18.3 

15.1 

14.2 

11.8 

11.2 

8 

11.0 

11.3 

11.0 

10.3 

10.8 

8 

17.8 

15.0 

14.8 

13.3 

13.3 

10 

12.1 

12.3 

12.0 

12.1 

13.3 

10 

17.4 

15.0 

15.8 

15.4 

15.3 

DIURNAL   VARIATIONS   OF   ELEMENTS 
TABLE  72 — (Continued) 


319 


5.     Vapor  Pressure  e 

6.     Electric  Potential  Fall. 

d 

Blue  Hill  (Summer) 

Kew 

Kremsmiinster 

Greenwich 

000 

200 

400 

1000 

Sum. 

Wint. 

Sum. 

Wint. 

Sum. 

Wint. 

0  A.M. 

11.07 

12.15 

8.78 

7.53 

0    A.M. 

+  3 

-  8 

-12 

-33 

+  6 

-  1 

2 

10.91 

11.40 

8.41 

6.80 

2 

-14 

-33 

-19 

-51 

+  1 

-  9 

4 

10.75 

11.16 

8.08 

6.44 

4 

-20 

-58 

-24 

-54 

-  3 

-13 

6 

10.88 

10.74 

8.15 

6.44 

6 

00 

-36 

+  2 

-23 

+  1 

-13 

8 

11.25 

10.75 

8.45    6.67 

8 

+23 

+  8 

+16 

+  4 

+  1 

-  6 

10 

11.14 

11.11 

8.92 

7.47 

10 

+14 

+34 

+13 

+22 

+  5 

+  5 

0  P.M. 

10.67 

11.62 

9.48 

8.19 

0  P.M. 

-10 

+10 

+  7 

+19 

-  3 

+  4 

2 

10.50 

11.53 

10.08 

8.84 

2 

-24 

-  6 

+  4 

+29 

-  9 

+  3 

4 

10.46 

11.12 

10.27 

9.02 

4 

-21 

+  8 

+  5 

+23 

-  8 

+  6 

6 

10.74 

10.85 

9.72 

8.81 

6 

-  2 

+25 

+  4 

+35 

-  3 

+  8 

8 

11.15 

10.79 

9.15 

8.31 

8 

+24 

+29 

+11 

+32 

+  1 

+  7 

10 

11.17 

11.38 

8.87 

7.81 

10 

+27 

+21 

-  7 

+  3 

+  8 

+  7 

2.  The  pressure  B  has  semidiurnal  waves  from  the  surface 
upward,  diminishing    to    extinction  on  the  3,000-meter    level. 
The  morning  crest  of  maximum  is  smaller  than  that  of  the 
afternoon.    There  is  not   the  least  evidence   that   the  semi- 
diurnal pressure  waves  embrace  an  oscillation  of  the  entire 
atmosphere,  as  Kelvin's  theory  of  the  forced  oscillations  demands, 
and  therefore  several  discussions  and  other  inferences  depending 
on  that  theory  are  really  without  proper  foundations. 

3.  The  vapor  pressure  is   subject   to   this  interchange   of 
periods.      At  Salton   Sea,  Southern   California,  Tower  No.   1 
was  located  in  the  desert,  1,500  feet  from  the  water,  and  the 
semidiurnal  period  is  clearly  denned  at  every  stage  from  the 
surface  to  40  feet.     Tower  No.  4  was  located  in  the  sea,  at  one 
mile  from  the  shore,  and  it  was  observed,  as  in  section  4,  that 
the  diurnal  vapor  pressure    ed    near  the  water  converts  itself 
into  a  semidiurnal  wave  within  40  feet  of  the  water.    At  Tower 
No.  1  the  diurnal  convection  could  not  obtain  vapor  from  the 
surface  sufficient  to  fill  up  the  diurnal  wave,  while  at  the  water 
this  deficiency  did  not  exist  near  the  surface,  in  consequence  of 
the  rapid  evaporation. 

Section  5,  for  the  vapor  pressure  in  the  free  air  at  considerable 
heights  above  the  surface  of  Blue  Hill,  shows  a  recombination 


320          TERRESTRIAL  AND  SOLAR  RELATIONS 

of  the  semidiurnal  waves  at  the  surface  into  diurnal  waves  at 
less  than  1,000  meters  above  the  surface.  Here  the  diurnal 
convection  carries  the  vapor  upward  to  cooler  strata  and  there 
concentrates  it  into  a  single  maximum  at  about  4  P.M.  The 
details  of  the  physical  conditions  of  these  periodic  interchanges 
must  be  left  for  more  minute  researches  into  the  prevailing  forces 
that  are  at  work. 

Section  6  gives  some  examples  of  the  well-known  change 
from  the  semidiurnal  waves  of  the  electric  potential  fall,  prevail- 
ing generally  in  the  summer  where  there  is  vigorous  convection 
in  the  lower  strata,  into  the  diurnal  or  approximately  diurnal 
wave  which  is  characteristic  of  the  winter  months.  Here  the 
positive  ions  in  the  atmosphere  appear  to  move  up  and  down, 
relatively  to  the  surface,  upward  in  the  convection  of  the  after- 
noon, thereby  diminishing  the  potential  gradient,  and  downward 
at  8  A.M.,  and  8  P.M.,  thereby  increasing  the  potential  gradients. 

Table  73  contains  a  collection  of  the  coefficient  of  dissipation 
af,  in  percentage  per  minute,  from  observations  made  throughout 
the  24  hours  at  Daroca,  Porta  Cceli,  Guelma,  Stations  of  the 
U.  S.  Eclipse  Expedition,  1905,  also  at  Bona,  and  on  the  U.  S.  S. 
Ccesar  during  the  voyage  across  the  Atlantic  Ocean. 

Daroca  is  on  the  Aragon  plateau;  Porta  Coeli  is  near  Val- 
encia; Guelma  is  in  the  interior  of  Algeria  and  Bona  is  the  port. 
The  observations  on  the  U.  S.  S.  Ccesar  were  made  during  the 
voyage  from  Gibraltar  to  the  United  States:  (i)  on  the  forward 
hatch,  and  (2)  under  the  shelter  of  a  large  iron  bulkhead.  The 
values  of  a' +  and  a'  _  at  Guelma  have  been  multiplied  by  the 
factor  ^2,  and  those  at  U.  S.  S.  Ccesar ,  bulkhead,  by  the  factor  2, 
in  order  to  reduce  them  to  the  scale  of  the  other  series  in 
taking  the  general  means.  These  data  are  plotted  in  Fig.  60  in 
percentage  per  minute  a' . 

Fig.  60  shows  that  the  diurnal  variation  of  the  dissipation 
coefficient  in  percentage  per  minute  has  a  maximum  at  about  3 
P.M.,  corresponding  with  the  vertical  convection,  and  secondary 
maxima  at  10  P.M.,  and  2  to  4  A.M.,  corresponding  with  the  two 
descending  branches.  The  minor  crests  occur  at  1  A.M.,  4  A.M., 
8  A.M.,  0  P.M.,  3  P.M.,  5  P.M.,  and  9  P.M.  Each  of  them  corresponds 


DIURNAL   VARIATIONS   OF   ELEMENTS 


321 


U 


UK 


1 


UK 


O 


OOOi-HOOOSOOO 


OOOOSO         O>OM«C500-#001010-<J<CO-<JI 
10  10  «0  <0          t-OOOOt-t-00<Ot>t-OOC-<0 


0  0  T}<  0  05  <N 

T*    10    l>    T}<    «>    «0 


«D  tO  ^  O  O  •«*•          00  «O  «O  «O  rj<  O 
C^OOkO^t~«D          t-t->-<t-kOOO 


cgoooor-iaseo       oooia>«Deo 

CJCO^O^^OCO         LClO^UdUD 


«otO(N<ocoeo 
o  eo  ^  eo  «o  TJI 


322 


TERRESTRIAL   AND    SOLAR  RELATIONS 


with  some  feature  of  the  diurnal  circulation,  as  it  transports  the 
ions  from  one  level  to  another  in  this  complex  system  of  local 


Oa.m.  2 


10         Op.m.  2 


10         0  a.m. 


1.60 


1.50 
1.40 


-CtA 


1.60 
1.50 
1.40 


3 


1 


FIG.  60.     Mean  diurnal  variation  of  the  coefficient  of  electrical  dissipa- 
tion in  percentage  per  minute. 


*1 

c 
AT         +1.0 

Temperature       o.O 

.m.2       4       6        8     10    Op.m.  2       4        6       8      10      \ 

Cordoba 

/ 

__5 

/ 

\ 

^ 

^~ 

\ 

2 

\ 

600  meter  level  -1.0 

m.m. 
AB           -0.50 
Pressure         o.OO 

\s 

4 

X^_ 

_s/ 

400-1000  meters 
Cordoba 

"  

' 

/• 

^ 

N 

~7 

\ 

/ 

+  0.50 

A                m.m. 
A0           -1.00 
Vapor  pressure    0.00 

^ 

"X 

/ 

\ 

Surface 
Salton  Sea 

^^^ 

„  

. 

7 



~  —  ^ 

/ 

\ 

/ 

\ 

^ 

-\ 

/ 

+1.00 

V 
AV               -25 
Electric  Potential  0.0 

_^--' 

^^» 

^ 

\ 

_-^i 

California 

Grenwich 
Paris 

s^^ 

X 

. 

/ 

\ 

/ 

\ 

. 

x 

+  25 

a' 

+  0.10 
Electric                   0 

"X. 

/ 

V 

/ 

Potsdam 
Potsdam 

,  

X. 

^~ 

1      • 

^^ 

/ 

> 

/ 

^ 

^^ 

Dissipation       -0.10 

^ 

^  —  ' 

Daroca 

FIG.  61.    Summary  of  the  various  semidiurnal  waves. 

circulations.     These   data   should   be   much   more   extensively 
studied. 

It  is  generally  found  that  the  coefficient  of  dissipation  varies 


DIURNAL   VARIATIONS   OF   MAGNETIC   FIELD  323 

as  follows:  (1)  Greater  in  clear  air  and  less  in  cloudy,  dusty  air; 
(2)  Greater  with  increase  of  the  wind  velocity;  (3)  Greater 
with  increase  of  the  temperature;  (4)  Greater  with  the  higher 
vapor  pressure.  We  may  finally  compare  the  diurnal  curves 
of  the  several  elements  with  the  temperature  waves  in  the  strata 
400-1,500  meters  above  the  surface. 

The  distribution  of  the  evaporation  in  the  soil  from  the  sur- 
face to  100  c.m.  has  been  carefully  worked  out  at  Cordoba,  with 
the  result  that  the  evaporation  from  the  water  table  in  the  soil 
takes  place  in  a  diurnal  curve-,  exactly  agreeing  with  that  of 
the  vapor  pressure.  It  is,  therefore,  thought  that  A  e,  A  F,  a' 
of  Fig.  61  should  be  inverted  and  referred  to  subsurface  evap- 
oration of  ground  water. 

The  Diurnal  Variations  of  the  Terrestrial  Magnetic  Field 

There  is  another  effect  of  the  diurnal  circulation  in  the  earth's 
atmosphere,  first,  in  the  generation  of  electric  currents,  and 

(-)  North  Pole 

'Day 

Night      /  \    I  Day 


Night 


(+)  South  Pole 
FIG.  62.     Scheme  of  the  diurnal  circulation  in  zones. 

secondarily,  in  the  induction  of  the  diurnal  magnetic  deflecting 
vectors  that  cause  the  variations  of  the  normal  magnetic  field. 


324 


TERRESTRIAL  AND    SOLAR   RELATIONS 


In  Bulletin  No.  21,  U.  S.  W.  B.,  1898,  were  published  the  results 
of  a  computation  on  the  observed  elements  H.  D.  V.  at  30 
stations. 

TABLE  74 
THE  MEAN  MAGNETIC  DEFLECTING  VECTORS  IN  FOUR  ZONES 


Arctic  Zone 

North  Temperate 

Tropic   Zone 

South  Temperate 

Zone 

Zone 

Hours  of  the 

Observations 

Mag.  Lat. 

Mag.  Lat. 

Mag.  Lat. 

Mag.  Lat. 

78°  to  62° 

61°  to  28° 

+  10°  to  - 

15° 

—  30°  to  —  55° 

Stations  7 

Stations  13 

Stations 

5 

Stations  5 

s        a         p 

s        a         p 

5         a 

f 

sa0 

Midnight 

60    -36°  345° 

15   -30°  111° 

20   -33° 

5° 

19    +27°  259° 

1  A.M. 

63    -44    355 

14   -35     109 

19    -32 

16 

19    +31     250 

2 

69   -43        5 

14    -32     102 

20   -36 

7 

17    +35     251 

3 

74   -44      16 

14   -33     108 

20    -42 

6 

18    +36     243 

4 

75    -42      25 

15    -35     112 

18    -34 

10 

20   +36    226 

5 

77    -42      30 

17    -33     110 

17    -37 

6 

21    +33     223 

6 

78    -40      32 

20    -31     112 

19    -36 

4 

24   +31     222 

7 

76    -40      36 

22-6     107 

21    -37 

339 

26    +24    235 

8 

65    -37      45 

25+3      99 

24    -30 

297 

28   +28    248 

9 

54    -18      68 

26    +24      66 

26    +23 

228 

28   +33     256 

10 

39    +31     117 

27   +37      49 

35    +25 

210 

26    -27     296 

11 

47    +44     195 

25   +38    312 

43    +22 

204 

25    -37    327 

Noon 

56    +43    200 

33   +35    287 

43    +30 

193 

28    -41       47 

1  P.M. 

64   +42    204 

32    +26    277 

40   +31 

163 

32    -36       53 

2 

73    +41     206 

29    +23    268 

34   +27 

156 

30    -35       72 

3 

83    +39     206 

25    -24    263 

17    +27 

121 

30    +30      82 

4 

89    +39     206 

22    -39     259 

16    -31 

40 

28    +30      85 

5 

87    +36    209 

19    -45    260 

16    -25 

18 

24    +39       84 

6 

78    +37     209 

18    -54    234 

19    -22 

12 

20   +40      84 

7 

62    +34    212 

17    -44     183 

21    -30 

3 

17    +45       78 

8 

54   +32     219 

16    -40     183 

23    -30 

2 

17    +46     108 

9 

51    +11     256 

14    -39     103 

23    -30 

358 

17    +47     283 

10 

50    -  6    279 

15    -36       96 

23    -28 

2 

18    +41     269 

11 

51    -37     336 

13    -33     105 

23    -28 

6 

18    +36    264 

s  =  the  vector  in  units  of  the  5th  decimal  (C.  G.  S.),  o.ooooi  dyne. 

a  =  the  vertical  angle,  positive  above  the  horizon 

P  =  the  azimuth  angle,  from  the  South  through  E.  N.  W. 

(740)  s  =  (dx*+dy*+  </z2)*,  o-  =  (dxz+  dy2^,  tana  =  ~    tan/?  =  ~ 


Table  74  contains  a  condensed  summary  of  these  vectors,  s,  a,  /3, 
in  the  four  principal  zones  (Fig.  62),  that  for  the  Antarctic  being 
omitted  for  lack  of  observations.  Fig.  63  contains  a  diagram 


DIURNAL   VARIATIONS    OF   MAGNETIC   FIELD 


325 


of  the  vectors,  which  illustrates  the  system  to  some  extent, 
though  they  can  be  properly  studied  and  appreciated  only  by 


Midnight 


6a.m. 


6  a.m 


6p.m. 


Midnight 

Arrows  with  — »  for  angles  above  the  horizon,  +  a. 
Arrows  with  >—  for  angles  below  the  horizon,  —  a. 

FIG.  63.  Scheme  of  the  directions  of  the  deflecting  forces  causing  the 
diurnal  variations  of  the  magnetic  field  in  five  principal  zones.  (Figure 
on  page  90,  Bulletin  No.  21,  U.  S.  W.  B.,  1898.) 

reference  to  the  original  30-inch  globe  model.     The  reader  is 
directed  to  Bulletin  No.  21  for  further  discussion  of  these  data. 


326  TERRESTRIAL  AND   SOLAR  RELATIONS 

This  magnetic  system  has  constituted  a  difficult  problem  for 
solution,  as  it  is  necessary  to  have  a  simple,  world-wide  cause 
capable  of  producing  these  diverse  effects. 

The  most  prominent  fact  is  the  inversion  of  vectors  as  be- 
tween the  two  hemispheres,  and  it  is  easy  to  show  that  the 
diurnal  convection  is  oppositely  directed  in  reference  to  the 
normal  magnetic  field,  positive  in  the  southern  hemisphere  and 
negative  in  the  northern  hemisphere. 

In  the  Tropic  zone  the  air  rises  nearly  vertically  by  day  and 
falls  by  night;  in  the  Temperate  zones  it  flows  toward  the  poles 
by  day,  and  toward  the  equator  by  night,  being  oppositely 
directed  in  each  hemisphere  relative  to  the  positive  direction 
of  the  magnetic  field;  in  the  Arctic  and  Antarctic  zones  the 
movement  is  upward  by  day  toward  the  sun  and  downward 
at  night.  These  five  zones  of  circulation  are  marked  off  from 
each  other  by  the  high-pressure  belts  in  latitudes  +  30°  and  —  30° 
and  by  the  low-pressure  belts  in  latitudes  +  66°  and  —  66°. 
The  zones  of  circulation  agree  with  the  zones  of  magnetic  vectors 
as  defined  in  1892. 

Fig.  64  contains  a  scheme  of  the  circulation  vectors  (black), 
and  the  magnetic  vectors  (dotted),  as  derived  from  the  two 
sources  indicated.  There  is  remarkable  agreement  so  far  as 
the  observational  data  extend,  and  the  corresponding  portions 
of  the  circulation  adopted  by  natural  inference  agree  with  the 
parts  that  are  known.  It  is  generally  true,  (l)  that  the  circula- 
tion vectors  and  the  magnetic  vectors  are  at  right  angles  to  each 
other,  and  (2)  that  the  turning  points  in  both  systems  coincide 
in  all  parts  of  the  five  zones.  The  conclusion  is  almost  imperative 
that  the  circulating  vectors,  through  the  generated  ions  in 
streams,  induce  the  observed  magnetic  deflecting  vectors. 
While  there  is  much  to  be  done  by  observations  fully  to  verify 
this  theory,  it  is  clear  that  the  main  features  of  both  the  systems 
are  in  remarkable  conformity  to  the  known  facts  of  the  observa- 
tions. The  horizontal  and  vertical  components  of  the  two  sets 
of  vectors  in  Fig.  64  should  be  united  in  one  set  of  spacial  vectors, 
in  order  that  this  system  may  be  properly  comprehended.  The 
evidence  is  very  strong  that  the  magnetic  variations  depend 


DIURNAL  VARIATIONS   OF  MAGNETIC  FIELD  327 

Mn      2        4        6       8       10   Noon    2        4       6       8        10     12 


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FIG.  64.  The  probable  types  of  the  diurnal  wind  vectors  as  generators 
of  the  diurnal  magnetic  deflecting  vectors.  Full-lined  vectors  =  the  elec- 
trical currents  in  the  streams  of  the  diurnal  convection  secondary  vectors. 
Dotted  vectors  =  the  induced  magnetic  deflecting  vectors  as  computed  from 
the  observations  and  given  in  the  U.  S.  Weather  Bureau  Bulletin  No.  21,  page 
87,  1897. 


328 


TERRESTRIAL   AND    SOLAR   RELATIONS 


upon  ionization  in  the  lower  strata,  and  not  upon  any  system  of 
ionization  currents  in  the  upper  strata,  as  has  been  claimed. 
Besides  the  vector  directions  of  the  magnetic  forces  we  may 
approximately  obtain  the  forces  by  the  formula, 
(741)     A#  =  4wen+u+, 

where  e  =  3.4  X  10~10,  n+  =  the  number  of  positive  ions  per 
cubic  centimeter,  and  u+  the  velocity  of  the  circulation  in  centi- 
meters per  second.  The  available  data  for  the  level  400  meters 
are  probably  approximately  as  follows,  in  the  South  Temperate 
zone: 

TABLE  75 

THE  DEFLECTING  MAGNETIC  VECTORS  AS  COMPUTED  AND  OBSERVED 
A  H  =  4  TT  e  n  u   in   C.    G.    S.    units. 


A.M 

Formula 

0 

2 

4 

6 

8 

10 

47T 

12  56 

e 

3.4X10~10 

n+ 
u  + 
AH 
Observed 

1045 
35 
.00016 
.00019 

1226 
38 
.00020 
.00017 

1374 
40 
.00023 
.00020 

1284 
36 
.00028 
.00024 

1245 
34 
.00018 
.00028 

1223 
46 
.00024 
.00026 

P.M 

. 

Formula 

0 

2 

4 

6 

8 

10 

47T 

e 

n  + 
u  + 
AH 
Observed 

1195 
75 
.00038 
.00028 

1232 
70 
.00037 
.00030 

1287 
55 
.00030 
.00028 

1225 
50 
.00026 
.00020 

1242 
45 
.00024 
.00017 

1274 
38 
.00021 
.00018 

The  number  of  ions  per  cubic  centimeter  was  obtained  at 
the  surface  at  Daroca  and  Guelma,  1905,  and  from  the  other 
available  published  data;  the  velocity  of  the  moving  medium 
in  cms/sec,  was  adopted  from  the  study  of  the  Argentine  data. 
The  results  are  so  far  in  harmony  with  the  observed  A  H  =  s  of 
Table  74,  South  Temperate  zone,  equivalent  approximately  to 


THE   APERIODIC   MAGNETIC  VECTORS  329 

5  of  the  North  Temperate  zone,  that  we  must  admit  that  there 
is  a  close  causal  connection.  The  magnitude  and  direction  of 
the  deflecting  magnetic  vectors  are  so  far  in  harmony  with  the 
convectional  vectors,  in  all  parts  of  the  earth,  that  the  subject 
will  deserve  to  be  further  studied,  especially  in  the  determina- 
tion of  the  wind  vectors  in  the  lower  strata  of  the  atmosphere. 

The  Aperiodic  Magnetic  Vectors  Along  the  Meridians 

According  to  Tables  66,  67,  and  Fig.  55,  there  are  two  principal 
regions  of  the  absorption  of  the  incoming  solar  radiation,  the 
cirrus  region  and  the  cumulus  region,  in  both  of  which  there 
is  transformation  of  energy  into  heat  or  into  electric  ions.  The 
consequences  of  such  ionization  have  been  studied  in  the  cumulus 
region,  in  the  induced  periodic  diurnal  or  low-level  variations 
of  the  magnetic  field.  It  remains  to  give  some  account  of  the 
effects  of  the  ionization  in  the  cirrus  region  upon  the  earth's 
normal  magnetic  field.  In  order  to  analyze  this  subject  the 
hourly  variations  are  eliminated  by  taking  the  mean  daily  values 
of  H  the  horizontal  force,  D  the  declination,  and  V  the  vertical 
force,  as  commonly  published.  As  an  example  of  the  world- 
wide correlation  of  these  daily  movements  of  the  magnetic  field 
the  horizontal  force  is  transcribed  in  scale  divisions,  or  units  of 
force,  for  Greenwich,  Toronto,  Singapore,  St.  Helena,  Cape  of 
Good  Hope,  Hobarton,  very  widely  separated  in  latitude  and 
longitude.  It  is  seen  that  substantially  the  same  sort  of  varia- 
tions, +  A  H,  —  A  H,  occur  nearly  simultaneously  all  over  the 
earth.  Similarly,  there  are  +  A  D,  -  A  D,  +  A  F,  -  A  V 
variations  occurring  from  day  to  day.  These  rectangular 
variations  must  first  be  all  transformed  to  C.  G.  S.  units  d  x, 
dy,  dz,  and  these  are  to  be  combined  in  polar  co-ordinates  s, 
a,  8,  which  give  the  magnetic  deflecting  vectors  disturbing  the 
normal  field  of  the  earth.  In  computing  A  H,  A  D,  A  F  from 
day  to  day,  since  there  is  an  incessant  secular  or  long-period 
variation  of  H,  D,  V,  it  is  necessary  to  secure  a  proper  base 
line,  with  appropriate  slope,  to  which  A  H,  AD,  A  F  may  be 
referred.  This  is  best  done  by  constructing  the  10-day  con- 


330 


TERRESTRIAL   AND    SOLAR   RELATIONS 


secutive  means,  H0)  D0,  V0  for  every  day  of  the  year,  so  that 
the  consecutive  mean  plus  the  variation  is  the  observed  value, 
H0  +  A  H  =  H ,  D0  +  A  D  =  D,  F0  +  A  V  =  V.  In  passing 
the  dates  of  excessive  magnetic  storms,  it  is  proper  to  substitute 
a  minimum  A  Hmin  =  0.00025  C.  G.  S.  It  has  been  proposed 
to  obtain  the  normal  field  by  taking  out  the  " quiet'7  days  for 


.0370 
Greenwich   .0360 
Abstract,  p.7  .0350 

560 
Toronto        550 
p.  395           540 

.1050 
Singapore    .1000 
p.  15         .0950 

60 
St.Helena        £8 
p.  81             56 

66 

Cape  Good  Hope  64 
p.  161            62 

120 
."Hobarton      118 
p.  371          116 

7 

+20 
+10 
Direct  Type       0 
-10 
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1 

8 

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5 

0 

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8 

9 

10 

11 

12 

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14 

15 

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17 

18 

19 

20 

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FIG.  65.  Example  of  the  variations  of  the  ^-component  of  the  magnetic 
field  in  all  latitudes,  and  showing  the  derived  normal  type  curve  in  the 
26.68-day  period  (direct),  beginning  August  3.58,  1845.  y  =  0.00001  C.  G.  S. 
unit. 

each  month,  and  computing  the  means  from  these  selected 
days.  Unfortunately  the  fact  that  a  day  is  "quiet"  does  not 
guarantee  that  the  day  is  near  the  normal,  because  "quiet" 
days  are  as  likely  to  run  on  one  side  of  the  normal  as  are  the 
rough  or  moderately  disturbed  days.  The  best  daily  and  monthly 
means  are  derived  by  taking  all  the  observations  as  they  occur, 


THE   APERIODIC  MAGNETIC   VECTORS 


331 


except  that  all  variations  greater  than  0.00025  C.  G.  S.  shall  be 
counted  at  that  value  for  the  sake  of  taking  out  the  consecutive 
means.  There  has  been  great  confusion  in  the  instrumental 
data,  in  the  manner  of  discussing  the  variations,  and  in  the 
interpretation  of  the  results. 

The  computed  magnetic  vectors  have  been  found  to  possess 


.0380 
Greenwich    .0370 
Abstract.p.7  .0360 

620 
Toronto         610 
p.  401           600 

.1050 
Singapore    .1000 
p.  18         .0950 

52 
St.Helena         50 
p.  87              48 

M 
Cape  Good  Hope  52 
p.  1C7             50 

114 

Hobarton       112 
p.  377           110 

y 

+  20 
+10 
Inverse  Type      0 
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FIG.  66.  Example  of  the  variations  of  the  /^-component  in  the  26.68-day 
period  (inverse),  beginning  November  18.30,  1845.  A  careful  study  of  these 
curves,  1843-1905,  shows  that  this  type  (reversed)  occurs  semiannually. 
Direct  type,  Feb.  1-April  20  and  July  15-Oct.  15;  inverse,  April  20-July  15 
and  Oct.  15-Feb.  1. 

interesting  and  important  characteristics.  Bulletin  No.  21, 
U.  S.  W.  B.,  1898,  contains  a  full  explanation  of  these  data,  to 
which  the  reader  is  referred.  (1)  These  vectors  of  deflection 
are  generally  closely  confined  to  the  magnetic  meridians,  and 
depend  chiefly  upon  A  H,  A  V.  According  to  the  latitude  of 


332 


TERRESTRIAL  AND   SOLAR  RELATIONS 


the  station  the  vector,  s,  a,  /3  has  well-defined  lengths  and  vertical 
angles,  similar  to  those  seen  in  Figs,  68,  69.  Corresponding 
with  the  waves  in  Figs.  65  66  these  vectors  point  first  south- 


FIG.  67.    Generation  of  the  ions  in  the  cirrus  and  cumulus  regions.     They 
flow  alternately  toward  the  north  and  south  poles. 


FIG.  68.    Closed  circuit  for  southward  magnetic  vectors.    The  +  ions  flow 
toward  the  north  pole. 

ward,  Fig.  68,  and  then  northward,  Fig.  69,  alternating  about 
every  three  days.  (2)  A  very  extensive  study  of  these  vectors  for 
the  years  1843-1910  shows  that  they  have  a  well-defined  period 
of  recurrence,  on  the  average  26.68  days  in  length,  and  witji  two 


THE  APERIODIC  MAGNETIC  VECTORS 


333 


types,  the  direct  as  in  Fig.  65,  and  the  inverse,  Fig.  66,  the 
relative  intensity  from  day  to  day  being  shown  in  the  lower 
section  of  each  figure.  The  recurrences  are  complicated  with 
many  irregularities,  but  the  periodic  action  is  unmistakable 
and  corresponds  with  the  synodic  period  of  the  rotation  of  the 
sun  on  its  axis,  as  observed  in  the  equatorial  zone.  The  inference 
follows  that  these  magnetic  meridian  deflecting  vectors  depend 
upon  certain  variations  in  the  solar  radiation,  distributed  in 


FIG.  69.     Closed  circuit  for  northward  magnetic  vectors.     The  +  ions  flow 
toward  the  south  pole. 


solar  longitudes  in  such  a  manner  that  certain  areas  of  the  solar 
surface  emit  stronger  radiations  than  do  others  in  different 
longitudes.  The  equatorial  period,  26.68  days,  is  exactly  the 
same  as  the  period  determined  from  numerous  direct  observations 
on  the  sun  spots,  the  faculae,  and  certain  spectrum  lines.  From 
a  least  square  solution  of  the  magnetic  data,  an  ephemeris  was 
constructed  on  the  period  26.679  days,  and  epoch,  June  13.72, 
1887. 

(3)  The  periodic  reversal  of  the  type  curve  occurs  in  semi- 
annual periods,  as  determined  by  the  records,  1841-1894.  Take 
the  successive  periods  by  years  and  match  the  type  curve  with  the 
observations  as  in  Figs.  65  and  66. 


334 


TERRESTRIAL  AND  SOLAR  RELATIONS 


TABLE  76 

SOLAR  MAGNETIC  EPHEMERIS,  PERIOD  26.679  DAYS,  EPOCH 
JUNE  13.72,  1887 


1840 

Jan.  16.87 

1870 

Jan.  24.05 

1900 

Jan.  5.55 

41 

Jan.  24.38 

71 

Jan.  5.88 

01 

Jan.  14.06 

42 

Jan.  6.21 

72 

Jan.  14.39 

02 

Jan.  22.57 

43 

Jan.  14.72 

73 

Jan.  21.90 

03 

Jan.  4.40 

44 

Jan.  23.23 

74 

Jan.  3.73 

04 

Jan.  12.91 

45 

Jan.  4.06 

75 

Jan.  12.24 

05 

Jan.  20.42 

46 

Jan.  12.57 

76 

Jan.  20.75 

06 

Jan.  2.25 

47 

Jan.  21.08 

77 

Jan.  1.59 

07 

Jan.  10.76 

48 

Jan.  2.91 

78 

Jan.  10.09 

08 

Jan.  19.27 

49 

Jan.  10.42 

79 

Jan.  18.60 

09 

Jan.  26.78 

1850 

Jan.  18.93 

1880 

Jan.  27.11 

1910 

Jan.  8.61 

51 

Jan.  27.44 

81 

Jan.  7.94 

11 

Jan.  17.12 

52 

Jan.  9.27 

82 

Jan.  16.45 

12 

Jan.  25.63 

53 

Jan.  16.78 

83 

Jan.  24.96 

13 

Jan.  6.46 

54 

Jan.  25.29 

84 

Jan.  6.79 

14 

Jan.  14.97 

55 

Jan.  7.12 

85 

Jan.  14.30 

15 

Jan.  23.48 

56 

Jan.  15.63 

86 

Jan.  22.81 

16 

Jan.  5.31 

57 

Jan.  23.14 

Epoch  87 

Jan.  4.64 

17 

Jan.  12.82 

58 

Jan.  4.97 

88 

Jan.  13  .  15 

18 

Jan.  21.33 

59 

Jan.  13.48 

89 

Jan.  20.66 

19 

Jan.  3.17 

1860 

Jan.  21.99 

1890 

Jan.  2.49 

1920 

Jan.  11.67 

61 

Jan.  2.82 

91 

Jan.  11.00 

21 

Jan.  19.18 

62 

Jan.  11.33 

92 

Jan.  19.51 

63 

Jan.  19.84 

93 

Jan.  27.02 

64 

Jan.  1.67 

94 

Jan.  8.85 

65 

Jan.  9.18 

95 

Jan.  17.36 

66 

Jan.  17.69 

96 

Jan.  25.87 

67 

Jan.  26.20 

97 

Jan.  6.70 

68 

Jan.  8.03 

98 

Jan.  15.21 

69 

Jan.  15.54 

99 

Jan.  23.72 

THE  SEMIANNUAL  REVERSAL  OF  THE  DIRECT  AND  INVERSE  TYPES 


Period 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

D 

17 

32 

42 

41 

18 

7 

13 

33 

43 

41 

30 

16 

10 

12 

I 

37 

22 

12 

13 

36 

47. 

41 

21 

11 

13 

24 

38 

44 

42 

Type 

/ 

D  £W.  D 

/  /max.  / 

D  £>max.  D   D 

/  /max.  / 

The  direct  type  prevails  annually,  February  1  to  April  20. 
The  direct  type  prevails  annually,  July  15  to  October  15. 
The  inverse  type  prevails  annually,  April  20  to  July  15. 


SYNCHRONOUS   ANNUAL  VARIATIONS  335 

The  inverse  type  prevails  annually,  October  15  to  February  1. 

These  facts  of  periodic  action  from  the  sun  in  the  equatorial 
period  of  26.68  days,  together  with  the  semiannual  inversion 
of  the  type,  indicate  that  the  problem  of  the  solar  radiation  at 
the  sun,  and  in  its  effects  throughout  the  earth's  atmosphere  is 
an  exceedingly  complex  phenomenon,  which  will  require  extensive 
researches  of  various  kinds. 

By  way  of  suggestion  it  may  be  seen  on  Fig.  67  that  if  the 
incoming  radiation  transforms  a  part  of  its  energy  in  the  cirrus 
region  into  positive  (+)  and  negative  (  — )  ions,  it  may  be 
supposed  that  they  seek  the  poles  of  the  earth  in  opposite  direc- 
tions, as  (+)  to  the  north  pole  and  the  (  — )  to  the  south  pole, 
completing  their  circuit  through  the  outer  shell  of  the  earth. 
This  generates  the  magnetic  vector  system  pointing  southward, 
and  the  corresponding  earth  electric  currents;  at  another  time  the 
(+)  ions  seek  the  south  pole  and  the  (  — )  ions  the  north  pole, 
thus  producing  the  northward  vectors,  and  the  corresponding 
earth  electric  currents.  This  reversal  of  direction  from  time 
to  time  depends  upon  the  physical  condition  of  the  atmosphere 
as  a  conducting  medium  for  the  ions,  its  congestion  of  ions,  its 
accumulation  of  ice  and  vapors,  producing  the  magnetic  vectors, 
auroras,  magnetic  storms,  electric  currents,  in  the  well-known 
conditions  as  observed.  The  energy  expended  at  the  earth 
is  that  transformed  from  the  solar  radiation;  it  is  inexhaustible 
in  amount,  and  depends  for  the  observed  aperiodic  irregularities 
upon  the  prevailing  states  of  the  solar  and  terrestrial  atmospheres. 

The  Synchronous  Annual  Variations  of  the  Solar  and  the  Terres- 
trial Elements 

The  possibility  of  a  scientific  forecast  of  the  type  of  weather 
likely  to  prevail  in  a  large  country  as  the  United  States  or 
Argentina,  whether  the  coming  year  is  to  be  rainy  and  cool,  or 
dry  and  warm,  depends  upon  the  establishment  of  the  following 
two  propositions:  (1)  The  radiation  output  of  the  sun  is  a 
variable  quantity,  as  4  or  5  per  cent,  each  side  of  the  mean; 
(2)  The  meteorological  elements,  temperature,  barometric  and 


336  TERRESTRIAL  AND   SOLAR  RELATIONS 

vapor  pressures,  and  the  precipitation  synchronize  with  the 
solar  changes  in  their  annual  variations.  The  evidence  at 
present  enables  us  to  affirm  that  both  are  true,  and  that  the 
synchronism  exists,  though  in  a  very  complex  form,  because  the 
prevailing  local  conditions  depend  primarily  upon  the  general 
circulation,  and  therefore  only  indirectly  upon  the  solar  varia- 
tions. It  is  not  possible  in  this  place  to  do  more  than  summarize 
the  general  principles  that  have  been  established  in  a  research 
extending  over  twenty  years,  and  embracing  the  available  solar 
and  terrestrial  data.  The  first  task  is  to  procure  homogeneous 
material  of  the  several  observed  quantities,  extending  over  a 
long  series  of  years,  sun-spot  frequencies,  solar -prominence 
frequencies,  amplitudes  of  the  terrestrial  magnetic  field,  baro- 
metric pressures  in  all  parts  of  the  world,  temperatures,  and 
vapor  pressures  in  all  countries,  precipitation  in  many  districts, 
direct  observations  of  the  solar  radiation  in  calories  per  square 
centimeter  per  minute.  Unfortunately  the  difficulties  of  secur- 
ing such  homogeneous  data  of  any  of  these  elements  is  greatly 
complicated  by  the  irregular  and  inconsistent  methods  that 
have  been  employed  by  meteorologists.  In  consequence  of  the 
necessity  of  substituting  a  few  selected  hours  of  observing  for 
the  twenty-four  hours  of  each  day,  it  is  necessary  to  reduce  the 
means  from  selected  hours  to  the  mean  of  twenty-four  hours,  which 
involves  a  long,  special  research  for  each  country.  The  selected 
hours  are  different  in  different  countries;  the  series  are  broken 
by  changes  in  the  selected  hours  in  consequence  of  some  admin- 
istrative requirement;  the  corrections  change  from  place  to 
place  when  the  same  hour  is  made  the  basis  of  the  work,  as 
where  the  75th  meridian  of  the  United  States  is  made  the  hour 
of  observing,  which  involves  a  range  of  three  hours  locally 
between  the  Atlantic  and  the  Pacific  States;  or  where  the 
Greenwich  noon  is  the  basis  of  simultaneous  world  observations, 
involving  variations  up  to  twelve  hours  in  local  conditions;  the 
altitudes  and  locations  of  the  instruments  in  great  cities  have 
been  not  infrequently  changed,  and  the  instrumental  equipment 
and  the  methods  of  computing  have  never  been  uniform  for 
the  long  series.  It  is  necessary  to  overcome  these  obstacles  by 


SYNCHRONOUS   ANNUAL   VARIATIONS 


337 


setting  aside  a  reasonable  number  of  permanent  stations  for 
long  series  of  fundamental  work  in  meteorology,  just  as  astrono- 
mers dedicate  certain  observatories  to  fundamental  star  places 
upon  which  the  National  Ephemerides  are  based.  Cordoba,  in 


1875 


1885 


1900 


1905 


£  ""  "^                                £- 

~~-^ 

/                   ._                    _J 

^ 

/                   S                    f_ 

^ 

11.1  year 

\                        2 

2                  *\    Z 

's 

^            £.        /          s-^^s^ 

^         ^         ^ 

"-   —  4              f-  -i  -  -  - 

7          /^  ~~  /  ^ 

'     \/       y        / 

_/                        r-                       '              ~f~  "^ 

C7     \^   52     ^ 

+400 

+  200 

z    V                  <c 

0    1                                       0 

!^/>        y^        _^       1 

^  i    'i  E  5  " 

r  7                   L 

Z.     ^^     a   1 

-0.50 

-0.25        / 

5  I—  Z              ^v^     \- 

T  "~    ~                   T  ~/    »  "^" 

Argentine             0.00 

L-           -X  ~  \-         4     7 

Barometric         +0.26      / 

Pressure               -     ~? 

/.                               s  _ 

+  0°50       -4 

" 

+o'.25       7 

n/N     /~\      ^_v      ^ 

J                                   Ji     X 

Centigrade         "O*25 

^                Z              ^ 

' 

T 

+100  /I 

f  r            'Sv        L 

"  x     \2.  £  \          'A 

o.i  / 

—  —  H                  A  Z  J  Z  _ 

PL        ^2        w          S 

/                      1                / 

\  y     v  / 

/                                                  / 

4.Yft  u 

y  y                  .~ 

+  05       ./ 

L       "u    6  3               /x 

^^          ^  ^     -/^ 

o.o     T 

t                               *— 

,                                  /       V 

'United  States     _  0  5 

T             \                7       ,    } 

7               ^  J 

Fahrenheit 

S  Z 

•* 

+  15.0 

+  12.0 

T 

+  9.0 

"               "   " 

+  6.0 

C-  V    -^ 

+  3.0 

.-^                ^/  V 

T  r 

United  States             o.O 

/    "I         \_\     /. 

/       ... 

Excess  Precipitation  _  3  0  —  ' 

r     r               j^     ^. 

r" 

-  6.0 

r  _, 

•j  7           7 

-9.0      _ 

^                        v/ 

-12.0 

\/ 

-15.0 

\i_             \T 

FIG.  70.     Synchronism  between  the  annual  variations  of  the  solar  and 
terrestrial  phenomena. 

Argentina,  is  such  a  first-class  meteorological  station,  because, 
since  1870,  the  instruments  have  had  the  same  natural  exposure, 
and  practically  the  same  apparatus  has  been  used  continuously, 
subjected  to  numerous  tests  for  normality.  There  is  no  station 


338 


TERRESTRIAL  AND    SOLAR   RELATIONS 


TABLE  77 

THE  SYNCHRONISM  BETWEEN  THE  SOLAR  AND  THE  TERRESTRIAL  ANNUAL 
VARIATIONS  OF  THE  METEOROLOGICAL  ELEMENTS 


Year 

Sun-spots 

Promi- 
nences 

Horizont  • 
Mag.  Am. 

Argentina 

United  States 

CM 

R 

CM 

R 

CM 

R 

AB 

AT 

A.e 

Free. 

&B 

AT 

Free. 

mm. 

°C 

mm. 

mm. 

inch 

°F. 

inch 

1872 

720 

+500 

1800 

+845 

2033 

+679 

-0.94 

+0.21 

+0.29 

-  76 

-  9.60 

73 

645 

+151 

1700 

+353 

2054 

-251 

—  .58 

+  .15 

+  .72 

+  32 

+  9.35 

74 

578 

-  43 

1569 

-154 

1894 

+168 

+  .05 

-  .58 

-  .48 

+  78 

-  7.46 

75 

364 

-159 

1188 

-332 

1630 

-  61 

+  .47 

+  .09 

-  .31 

-  50 

-  0.89 

76 

213 

-  77 

865 

+  11 

1681 

-358 

+  .04 

+  .04 

+  .26 

+  94 

-  0.99 

77 

120 

+  27 

695 

+  47 

1715 

-320 

-  .52 

+  .72 

+  .70 

-104 

+  2.65 

78 

177 

-136 

791 

-353 

1948 

+110 

+  .25 

-  .62 

+  .20 

+252 

-0.021 

+0.6 

+22.11 

79 

260 

-188 

1015 

-454 

2256 

-  23 

+  .33 

-  .14 

-  .51 

-327 

+  .005 

+0.4 

-15.18 

1880 

373 

+  14 

1332 

+  6 

2729 

+  1 

-  .26 

-  .22 

-  .05 

-  69 

+  .023 

-1.4 

-  6.15 

81 

518 

+133 

1064 

+394 

2946 

-  80 

-  .41 

+  .37 

+  .46 

+107 

-  .010 

+0.4 

+  4.47 

82 

656 

+  60 

2063 

+264 

8144 

+613 

+  .22 

-  .18 

-  .49 

-108 

+  .003 

-0.2 

+  0.60 

83 

704 

+  61 

2252 

-456 

3193 

-  47 

+  .20 

-  .01 

+  -17 

+  73 

+  .021 

-0.3 

-  5.39 

84 

634 

+127 

2170 

+686 

3204 

+  19 

-  .18 

+  .37 

+  .36 

-  22 

-  .024 

-0.9 

+18.36 

85 

523 

+103 

2080 

+204 

2920 

+  53 

+  .14 

-  .48 

-  .19 

+174 

-  .011 

+1.6 

-  0.81 

86 

386 

-  81 

2098 

-513 

2779 

+142 

+  .33 

-  .25 

-  .36 

-  87 

+  .009 

-0.3 

-14.84 

87 

249 

-  92 

1671 

+206 

2569 

-233 

-  .17 

+  .35 

-  .11 

-204 

+  .010 

-0.6 

-  5.76 

88 

140 

-  59 

1352 

+535 

2298 

+144 

-  .43 

+  .54 

+  .48 

+  83 

-  .014 

+0.6 

+  7.59 

89 

165 

-  90 

1402 

-678 

2220 

-  48 

+  .11 

-  .45 

+  .15 

+217 

-  .009 

+0.6 

-  0.31 

1890 

309 

-224 

1521 

-836 

2473 

-854 

+  .50 

-  .26 

-  .77 

-186 

+  .016 

-0.6 

—  0.44 

91 

496 

-  69 

1592 

+244 

2613 

-  82 

-  .57 

+  .29 

+  .49 

+108 

-  .002 

+0.3 

-  0.51 

92 

669 

+207 

1740 

+732 

2795 

+804 

+  .04 

-  .10 

+  .08 

-  31 

-  .003 

+0.4 

+  3.69 

93 

805 

+214 

1931 

+310 

2993 

+151 

+  .65 

-  .42 

-1.03 

-185 

+  .003 

-1.0 

+  0.33 

94 

820 

+116 

1806 

-339 

3073 

+  7 

+  .05 

-  .43 

-  .29 

+  60 

+  .006 

-0.5 

-  5.31 

95 

708 

+  60 

1534 

+106 

2806 

-195 

-  .47 

+  .48 

+  .15 

+  35 

.000 

0.0 

-  7.48 

96 

568 

-  67 

1230 

-  192650 

+280 

-  .08 

+  .91 

+  .83 

+  21 

-  .007 

+0.7 

+  2.15 

97 

410 

-  95 

1037 

+  76 

2441 

-177 

+  .52 

-  .14 

+  .09 

-  17 

+  .005 

+0.2 

+  8.38 

98 

279 

+  42 

817 

-  96 

2236 

+130 

+  .09 

-1.10 

-  .91 

-112 

+  .001 

-0.4 

-  5.39 

99 

185 

-  40 

604 

-106 

1925 

+107 

-  .45 

+  .21 

+  .28 

+129 

-  .003 

-0.5 

-  2.80 

1930 

135 

-  21 

398 

+145 

1765 

-176 

-  .25 

+  .59 

+  .61 

+169 

-  .002 

+1.0 

+  6.63 

01 

129 

-  96 

295 

-150 

1717 

-343 

+  .11 

-  .04 

-  .67 

-211 

+  .008 

+0.3 

-  7.98 

02 

201 

-140 

305 

-222 

1853 

-389 

-  .44 

+  .16 

+  .11 

-134 

-  .017 

-0.7 

+  2.16 

03 

330 

-  38 

300 

-  96 

1896 

+230 

+  .22 

-  .32 

+  .17 

+  76 

+  .014 

-0.5 

-  0.85 

04 

453 

+  51 

400 

+150 

2000 

-213 

+  .01 

-  .03J+  .06 

+229 

+  .008 

+0.7 

-11.53 

05 

550 

+212 

600 

-  81 

2180 

+  14 

-  .08 

-1  .19  +  .12 

+  41 

-  .009 

0.0 

+  7.81 

06 

650 

—  4 

—  .17 

-l_  74'  —  01 

—  197 

+19.11 

07 

_  .08  —  .02;  +  .15 

-  fi 

+  0.26 

08 

+  .41  —  .26  —  .17  +  26 

09 

+  .20  -  .01  -  .47!  -115 



1910 

—  .16  +  .45  —  .20  —  66 

11 

-  .11|-  .46  -  .07  +178 

CM  =  consecutive  mean;   R  =  residual;    C  M  +  R  =  Observed. 
A  B  =  variation  of  barometric  pressure;  A  T  =  variation  of  temperature. 
A  e  =  variation  of  the  vapor  pressure;  Prec.  =  excess  of  precipitation. 

in  the  United  States  that  compares  with  it,  because  of  changes 
of  one  kind  or  another  in  the  instrumental  conditions  or  the 
hours  of  observing.  The  writer  spent  many  years  in  adjusting 
the  imperfect  observations  in  the  United  States,  and  finally 


SYNCHRONOUS   ANNUAL   VARIATIONS  339 

produced  a  set  of  series  of  Pressure,  Temperature,  Vapor  Pres- 
sure, and  Precipitation,  that  are  fairly  homogeneous,  and  form 
the  fundamental  basis  from  which  the  annual  variations  may  be 
computed.  Similar  reductions  to  homogeneous  data  are  being 
made  in  Argentina,  and  in  other  countries,  and  in  time  it  is 
hoped  that  world-wide  comparable  series  of  reduced  observa- 
tions may  be  made  accessible  to  the  scientific  public. 

Table  77  and  Fig.  70  contain  a  series  of  examples  of  the  results 
of  such  a  comparison  of  the  solar  and  terrestrial  annual  data, 
enough  to  give  the  reader  a  fair  idea  of  the  possibilities  of  this 
important  subject.  The  sun-spots  are  from  Wolfer's  data,  and 
the  consecutive  means  C.  M.  added  to  the  residuals  R  produce  the 
observed  annual  means  O,  C.  M.  +  R  =  O;  the  solar  prominence 
frequencies  are  from  the  data  of  the  Italian  observers;  the 
amplitudes  of  the  horizontal  magnetic  force  were  compiled  from 
several  European  observatories:  the  Argentine  Meteorologica 
can  be  found  in  Bulletin  No.  1,  Oficina  Meteorologica  Argentina, 
1911;  the  United  States  data  may  be  found  in  the  Barometry 
Report,  U.  S.  W.  B.,  1902;  Temperatures  and  Vapor  Pressures 
in  Bulletin  S,  1909;  Temperature  and  Precipitation  Normals, 
Bulletin  R,  1908;  Temperature  Departures,  Bulletin  U,  1911; 
Climatological  Summary  in  106  sections,  Bulletin  W,  1912,  all 
prepared  under  the  writer's  supervision.  The  consecutive 
means  represent  a  long  periodic  cycle  averaging  11.1  years  in 
duration,  but  very  irregular  in  length,  as  from  8  years  to  14  years 
between  certain  maxima;  the  residuals  represent  a  short  periodic 
cycle  averaging  3.75  years,  but  ranging  between  3  years  and  5 
years.  The  first  curves  of  Fig.  70  represent  the  11.1  years  and 
the  3.75  years  cycles  between  1875  and  1905.  Following  them 
are  several  curves  for  Argentina  and  the  United  States  in  the 
short  period  cycle.  The  synchronism  in  the  short  period  is 
pronounced,  in  spite  of  certain  irregularities,  demonstrating 
the  general  fact  that  the  sun  has  a  variable  output  of  radiation 
which  persistently  modifies  the  earth's  circulation  and  climatic 
conditions.  The  barometric  pressures  and  temperatures  were 
studied  in  all  parts  of  the  world  and  the  result  summarized  in 
Monthly  Weather  Review,  October  and  November,  1903. 


340  TERRESTRIAL  AND  SOLAR  RELATIONS 

Barometric  Pressure.  The  net  work  of  barometric  pressures 
for  the  world,  taken  for  the  annual  variations,  shows  that  the 
stations  must  be  divided  into  two  classes:  (1)  Those  where  the 
synchronism  is  direct  between  the  pressure  and  the  prominences, 
as  surrounding  the  Indian  Ocean,  and  those  where  the  synchron- 
ism is  inverse  as  in  North  and  South  America.  Under  the 
external  impulse  from  the  sun  an  increase  of  the  annual  radiation 
accelerates  the  general  circulation  in  such  a  way  that  the  pressure 
is  simultaneously  higher  in  certain  large  regions  and  lower  in 
others.  This  is  due  to  the  fact  that  the  total  pressure  of  the 
earth's  atmosphere  is  an  invariable  constant,  so  that  if  the 
pressure  in  one  region  is  relatively  high,  that  in  another  region 
is  relatively  low  at  the  same  time.  The  wandering  cyclones  and 
anticyclones,  added  to  the  more  permanent  centers  of  high  and 
low  pressures,  should  sum  up  to  the  same  constant  for  the 
world.  The  oscillation  of  regional  pressures  is,  therefore,  a 
fundamental  fact  leading  to  an  extensive  study  of  the  pressure 
conditions  in  various  localities. 

The  Temperatures.  Similar  studies  of  the  annual  tempera- 
tures divide  the  stations  into  two  groups,  (l)  Those  in  the 
Tropics  with  direct  synchronism,  (2)  those  in  the  Temperate 
zones,  on  the  poleward  side  of  the  high-pressure  belt,  with  inverse 
synchronism.  There  are  many  places  of  mixture  or  disintegrated 
effects  which  it  is  still  difficult  to  classify.  An  increase  of  solar 
radiation  increases  the  vertical  convection  of  the  Tropics,  with 
increase  of  the  surface  temperature;  this  is  followed  by  an  in- 
crease of  downflow  in  the  Temperate  zones,  with  an  extension 
of  the  high  areas  and  cooler  temperatures.  The  temperature 
integral  of  the  entire  earth's  atmosphere  must  be  nearly  a  con- 
stant, or  else  the  earth's  rotation  period  of  twenty-four  hours 
would  indicate  variations  of  an  astronomical  value,  which  have 
never  been  detected  in  the  observations. 

Precipitation.  The  changes  in  the  general  and  the  local  cir- 
culations, depending  upon  the  solar  variations,  carry  with  them 
the  rain-bearing  currents,  as  from  the  oceans  to  the  continents, 
and  thence  the  annual  amounts  of  the  precipitation  in  the  regions 
concerned.  There  are  great  irregularities  in  these  precipitations 


SYNCHRONOUS   ANNUAL   VARIATIONS  341 

from  one  region  to  another,  from  one  year  to  another,  and  for 
the  same  station.  The  results  from  Argentina  and  the  United 
States  indicate  clearly  that  the  precipitation  synchronizes  with 
the  solar  variations,  and  that  the  variations  are  of  large  amounts, 
ranging  through  400-500  millimeters  in  Argentina  at  the  same 
station,  and  several  inches  in  the  United  States. 

Partial  Formations.  Fig.  70  shows  that  the  annual  crests 
in  certain  elements  occasionally  fail  to  form  completely  in  the 
3.75-year  period,  and  for  this  cause  irregularities  appear  in  the 
series  of  curves.  It  is  easy  to  see  how  this  may  occur  in  many 
cases  by  a  sort  of  self-contradiction  in  natural  causes  and  effects. 
Thus,  if  in  a  certain  region  the  excess  of  solar  radiation  of  the 
Tropics  has  produced  higher  temperatures,  this  has  resulted  in 
spreading  a  rain  and  cloud  sheet  over  another  region  at  a  distance 
from  it,  both  due  to  the  same  cause.  This  very  cloud  sheet, 
however,  acts  as  a  screen  upon  the  surface  temperatures,  so 
that  lower  local  temperatures  are  registered  at  the  surface,  while 
they  are  really  higher  above  the  cloud  sheet.  The  rain  currents 
may  precipitate  so  much  aqueous  vapor  on  one  side  of  a  mountain 
range  that  the  overflow  on  the  other  side  is  dryer  than  usual,  so 
as  to  give  opposite  effects  for  the  same  efficient  increase  of 
circulation,  excess  in  one  region,  and  defect  in  another  region. 
The  observations  of  the  solar  prominences  depend  upon  the 
number  of  clear  days  per  month.  Hence,  an  increase  in  solar 
radiation,  following  an  increase  in  the  frequency  of  the  promi- 
nences, may  locally  produce  a  'cloud  sheet,  and  hence  a  lower 
annual  count  in  the  number  of  the  prominences.  It  is  quite 
irrelevant  to  attempt  to  discredit  the  facts  of  synchronism,  by 
presenting  irregularities  or  inconsistencies  in  certain  localities, 
unless  the  trouble  is  taken  to  understand  the  full  series  of  causes 
and  effects  between  solar  action  and  final  local  conditions. 
Since  opposite  results,  inversions  of  effects,  are  inevitable  in 
terrestrial  meteorology,  from  the  same  solar  cause,  it  will  be 
necessary  to  study  carefully  the  history  of  each  region,  before 
attempting  to  arrive  at  any  conclusions. 

The  magnetic  field  presents  similar  synchronous  variations, 
as  may  be  seen  by  plotting  the  amplitude  curves.  This  element 


342          TERRESTRIAL  AND  SOLAR  RELATIONS 

is  very  sensitive  to  many  radiation  and  ionization  influences, 
and  it  is  our  purpose  to  pursue  the  research  into  the  function 
connecting  these  several  elements. 

The  radiation  in  calories  per  square  centimeter  per  second 
does  not  yet  present  annual  variations  which  seem  to  be  reliable. 
The  cause  of  this  result  is  seen  in  the  section  on  radiation,  and 
may  be  verified  by  studying  the  divergent  annual  values  on 
Table  62. 

The  possibility  of  annual  forecasts  of  the  weather  conditions 
is  being  tested  in  Argentina  by  projecting  forward  the  normal 
3.75-year  curve  from  1911  to  1915.  The  results  for  1911,  1912, 
and  1913  are  entirely  successful,  the  precipitation  being  quite  the 
same  as  indicated  in  Bulletin  No.  1,  0.  M.  A.  It  is  certainly 
possible  to  make  similar  forecasts  for  the  United  States  as  to 
precipitation  in  different  districts,  wherever  the  sequence  of 
the  rainfall  in  each  district  is  studied  in  relation  to  the  funda- 
mental solar  3.75-year  period.  Compare  Abstract  No.  3,  U.  S. 
W.  B.,  1909.  with  the  data  of  Fig.  70. 

The  Aqueous  Vapor  in  the  Atmosphere 

It  is  evident  from  the  discussions  on  radiation,  on  cloud 
formation  and  precipitation,  and  on  evaporation  of  aqueous  vapor 
from  areas  of  water,  as  in  lakes  and  oceans,  that  the  presence  of 
aqueous  vapor  in  the  atmosphere  is  of  primary  significance. 
We  can  compute  the  number  of  grams  of  aqueous  vapor  per  cubic 
meter  of  air,  or  per  kilogram  of  air,  according  to  convenience. 

Grams  of  Aqueous  Vapor  in  1  Cubic  Meter  of  Saturated  Air 
(742)     ^  = 


The  full  form  most  used  at  all  elevations  above  the  sea  level. 

By  substituting  the  observed  vapor  pressure  e,  the  tempera- 
ture 7",  the  barometric  pressure  B,  at  any  other  point,  the 
corresponding  /*  can  be  computed.  Extensive  tables  have  been 


AQUEOUS   VAPOR   IN   THE   ATMOSPHERE  343 

prepared  for  //  where  T  ranges  from  —  50°  C.  to  +50°  C.,  and 
B  from  800  mm.  to  20  mm. 

Grams  of  Aqueous  Vapor  in  One  Kilogram  of  Saturated  Air. 

(743)  ft  =  0.622  ^  +  0.235  ^ 

eQ  is  the  saturated  vapor  pressure  for  temperatures  ranging 
from  -  50°  C.  to  +  50°  C. 

When  the  air  is  not  saturated  the  following  formula  serves: 
eQ  =  the  saturated  vapor  pressure,  /  =  the  dry-bulb  temperature, 
/i  =  the  wet-bulb  temperature. 

Vapor  Pressure  in  Millimeters  when  the  Air  is  not  Saturated 

(744)  e  =  eQ  -  0.00066  B  (t  -  tj  (l  +  g^) . 


Tables  applicable  to  practical  work  may  be  found  in  Bulletin. 
No.  2,  Oficina  Meteorologica  Argentina,  1912. 

In  the  free  air  the  aqueous  vapor  is  distributed  approximately 
by  Harm's  formula, 

h 
(745)     e  =  e,  10    6517, 

where  h  is  the  height  in  meters. 

The  Laws  of  the  Evaporation  of  Water  from  Lakes,  Pans,  and 
Soils  with  Plants 

The  subject  of  the  evaporation  of  water  has  been  very  ex- 
tensively studied,  and  there  is  a  large  literature  on  the  results. 
These,  however,  are  unsatisfactory  as  concerns  the  terms  and 
the  coefficients  of  the  proposed  formulas.  Another  research 
was  undertaken  by  the  writer  in  1907  for  the  U.  S.  Weather 
Bureau,  at  Reno,  Nevada,  where  the  proper  type  of  formula  was 
determined;  it  was  continued  in  1908  at  Indio  and  Mecca,  So. 
California,  and  at  the  Salton  Sea,  1909,  1910,  in  co-operation 
with  numerous  stations  in  various  parts  of  the  United  States, 
during  which  the  coefficients  were  approximately  computed;  the 


344  TERRESTRIAL   AND    SOLAR   RELATIONS 

work  was  continued  in  1911,  1912,  at  Cordoba,  Argentina,  and 
extended  to  include  evaporation  from  soils,  and  soils  with  plants 
of  different  kinds,  and  the  final  coefficients  with  the  necessary 
working  tables  for  the  computations  were  constructed.  The 
results  of  this  work  are  summarized  in  Bulletin  No.  2,  Argentine 
Meteorological  Office,  1912.  Several  special  pieces  of  apparatus 
have  been  invented:  Bigelow's  micrometer  hook  gage  for  measur- 
ing the  water  height,  Bigelow's  dial  gage  for  measuring  the  water 
height  in  soil  tanks,  Wilcken's  self -registering  apparatus  for 
continuous  records  of  every  position  of  the  water  surface.  The 
principal  difficulty  in  arriving  at  conclusions  has  been  due  to  the 
necessity  of  using  pans  for  evaporation,  in  which  case  the  wind 
in  blowing  over  the  pan  greatly  complicates  the  action  of  the 
evaporation.  Pans  of  different  sizes  in  the  same  wind  evaporate 
different  amounts  during  the  same  interval  of  time,  because  the 
wind  carries  away  the  evaporated  vapor  at  different  rates, 
according  to  the  size  of  the  pan,  and  thus  produces  a  varying 
mixture  of  dry  air  and  vapor.  A  large  body  of  water  in  a  wind, 
and  a  small  pan  in  a  calm,  produce  the  same  effect  as  an  evaporat- 
ing medium,  because  the  vapor  is  actually  the  same  in  density 
near  the  water  on  a  lake  in  a  wind,  which  merely  transports 
it  from  place  to  place  without  really  removing  it,  as  in  a  calm 
air  over  a  small  pan.  The  result  is  that  lakes  evaporate  only  at 
about  two-thirds  the  rate  from  pans  near  by  in  moderate  winds. 
In  certain  places  it  was  found  that  a  small  pan  evaporates  three 
times  as  much  water  as  does  a  lake  in  the  neighborhood.  For 
example,  there  were  three  towers  built  in  the  Salton  Sea,  No.  2 
near  the  shore,  No.  3  about  half-mile  from  No.  2,  and  No.  4 

TABLE  78 
EXAMPLES  OF  THE  ANNUAL  EVAPORATION  AT  THE  SALTON  SEA 

Tower  No.  1.  Pan  (5),  40  feet  above  the  desert,  195  inches. 

Tower  No.  1.  Pan  (1),  on  the  ground  of  the  desert,  165 

Tower  No.  2.  Pan  (5),  40  feet  above  the  water,  138 

Tower  No.  2.  Pan  (1),    2  feet  above  the  water,  109 

Tower  No.  4.  Pan  (5),  40  feet  above  the  water,  140 

Tower  No.  4.  Pan  (1),    2  feet  above  the  water,  106 

The  evaporation  from  the  Salton  Sea  itself,  72 


FORMULA   OF   EVAPORATION 


345 


about  one  mile  from  the  shore,  while  No.  1  was  1,500  feet  inland 
from  the  sea  in  the  desert.  These  towers  carried  pans  near 
the  surface  of  the  water  and  at  every  10  feet  up  to  40  feet  above 
the  water.  The  evaporation  for  a  year  was  as  follows  at  several 
pans,  as  summarized  in  Table  78. 

The  evaporation  was  registered  at  other  stations  from  pans 
of  different  sizes  on  the  ground,  and  on  a  stand  10  feet  high,  of 
which  annual  examples  follow. 

EVAPORATION  AS  RECORDED  IN  SEVERAL  PLACES 


Station 

Indio 

Mecca 

Brawley 

Mammoth 

Height 
Size  of  pan 
Evaporation 

Ground            10  feet 
6  feet               2  feet 
119                 200 

Ground        10  feet 
6  feet           2  feet 
108             170 

Ground        10  feet 
6  feet           2  feet 
104             164 

Ground       10  feet 
6  feet           2  feet 
126            179 

Station 

N.  Yakima 

Cincinnati 

Birmingham 

Lake 
Tahoe 

Lake 
Kechess 

Height 
Size  of  pan 
Evaporation 

Ground            10  feet 
4  feet               3  feet 
68                     86 

Water          10  feet 
4  feet           3  feet 
46                 62 

Water          10  feet 
4  feet           2  feet 
51                 64 

2  feet 
4  feet 
42 

10  feet 
3  feet 
33 

The  formulas  that  have  been  found  to  be  adequate  to  follow 

the  course  of  evaporation  in  all  climates,  that  is,  in  all  conditions 

of  temperature,  vapor  pressure,  and  wind  velocity,  are  as  follows : 

Hours  of  observation  for  4-hour  intervals  (2,  6,  10)  A.M. 

(2,  6,  10)  P.M. 

t  =  the  temperature  of  the  dry  bulb  on  the  whirling  psy- 

chrometer  as  usually  employed. 
ti  =  the  temperature  of  the  wet-bulb  thermometer. 
ed  =  the  computed  vapor  pressure  at  the  dew  point  d. 
S  =  the  temperature  of  the  water  surface. 
es  =  the  computed  vapor  pressure  at  saturation  S. 

de 

-rz  =  the  rate  of  change  of  the  vapor  pressure  with  the  tempera- 

d  o 

ture  change  of  the  water. 

w  =  the  velocity  of  the  wind  in  kilometers  per  hour,  derived 
from  the  successive  anemometer  readings. 


346  TERRESTRIAL  AND   SOLAR  RELATIONS 

Formula  of  Evaporation  from  Large  Water  Surfaces 
(746)      -T  =  °-0230  ~  ^  +  °'084  ™);    (ArSentine 


anemometer)  . 

Formula  of  Evaporation  from  Pans  of  Different  Areas 
=  0-0230  F  „       -     l   1  +  0.084  w). 


F  (w)  =  a  factor  depending  on  the  area  of  the  pan,  which 
varies  with  the  wind  velocity  up  to  about  10  kilometers  per 
hour.  FI  (w)  applies  to  the  Dines'  system  of  wind  velocities, 
used  by  the  Argentine  Meteorological  Office,  and  a  pan  of  1.0 
meter2  area;  F2  (w)  to  the  same  wind  system  and  a  0.5  m2  area 
pan;  F»  (w)  to  the  wind  system  used  in  the  United  States,  where- 
in the  same  wind  velocity  is  recorded  higher  in  the  ratio  1.21 
to  1.00,  and  a  1.0  m2  pan;  F4  (w)  to  the  U.  S.  wind  system  and  a 
pan  1.17  m2  area  or  4  feet  in  diameter;  F5  (w)  to  the  U.  S.  wind 
system  and  a  pan  0.29  m2  area,  or  2  feet  in  diameter.  There  are 
two  wind  systems  in  use:  (1)  that  based  upon  the  Dines' 
pressure-velocity,  and  (2)  that  based  upon  the  whirling  machine 
velocities.  Thus  the  anemometers  by  Casella,  Negretti,  and 
Zambra,  U.  S.  Weather  Bureau  Freiz,  Richard,  are  approxi- 
mately in  agreement  together,  but  they  are  about  20  per  cent 
higher  than  the  Dines,  Hess  of  the  Oficina  Meteorologica 
Argentina,  Munro,  and  Tschau  system  of  anemometers.  Mar- 
vin's table  of  corrections  to  the  Robinson  anemometer  gives 
about  20  per  cent  correction  to  reduce  from  the  indicated  to 
the  true  wind  velocity,  Monthly  Weather  Review,  October,  1906, 
Table  64,  so  that  the  first  group  becomes  equivalent  to  the 
second  group  after  making  this  reduction.  Unfortunately,  it 
is  customary  to  omit  these  reductions,  so  that  the  published  wind 
velocities  of  the  United  States,  and  other  countries  using  the 
above-mentioned  anemometers,  are  about  20  per  cent  too  great. 
It  is  indispensable  in  evaporation  reductions  that  the  coefficients 
should  be  adjusted  to  correct  wind  velocities.  For  this  purpose 
the  following  factors  F  (w)  are  introduced  into  the  working 
Tables: 


FORMULA  OF  EVAPORATION 


347 


TABLE  79 

THE  FACTORS  F  (w)  FOR  ADJUSTING  THE  EFFECTS  OF  THE  WIND  VELOCITIES 
FOR  PANS  OF  DIFFERENT  AREAS 


System 

Argentina 

United  States 

Wind 

1.0m2 

0.5  m2 

1.0  m2 

1.17  m2 

0.29  m2 

Velocity 

Mw) 

F»(w) 

F,(w) 

F4(w) 

Ft» 

w  =    0 

1.000 

1.000 

.000 

1.000 

1.000 

1 

1.150 

1.148 

.120 

1.150 

1.160 

2 

1.265 

1.274 

.240 

1.212 

1.290 

3 

.376 

1.392 

.320 

1.289 

1.410 

4 

.463 

1.493 

.400 

1.367 

1.520 

5 

.542 

1.592 

.480 

1.433 

1.630 

6 

.600 

1.667 

.540 

1.480 

1.710 

7 

.617 

1.712 

.590 

1.523 

1.760 

8 

.627 

1.746 

.615 

1.542 

1.810 

9 

.629 

1.762 

.623 

1.552 

1.830 

10 

1.629 

.777 

1.629 

1.561 

1.840 

15 

1.629 

.782 

1.629 

1.561 

1.850 

2Q 

1.629 

.782 

1.629 

1.561 

1.850 

25 

1.629 

.782 

1.629 

1.561 

1.850 

30 

1.629 

.782 

1.629 

1.561 

1.850 

The  complete  tables  for  evaporation  computations  may  be 
found  in  Bulletin  No.  2,  Argentine  Meteorological  Office,  1912. 

It  has  been  found  that  about  90  per  cent  of  the  computed 
results  are  less  than  0.30  mm.  from  those  as  observed.  This 
difference  includes  the  errors  of  measurement  as  well  as  of 
computation.  The  computed  difference  from  the  observed 
amounts  for  entire  months  in  Cordoba  is  about  4  millimeters, 
and  the  total  difference  for  the  year  on  one  pan  was  —  3  milli- 
meters, and  on  another  pan  —  10  millimeters,  the  total  in  the 
first  case  being  1,091  millimeters,  and  in  the  second  case  1,945 
millimeters.  Our  experience  leads  us  to  conclude  that  pans 
need  not  be  employed  in  work  on  evaporation,  but  that  compu- 
tations are  quite  as  accurate  provided  observations  of  the  water 
temperature  S,  the  vapor  pressure  of  the  air  edj  and  the  velocity 
of  the  wind  w  are  made.  As  it  is  impossible  to  float  pans  on 
large  bodies  of  water,  lakes  and  reservoirs,  except  under  restricted 


348          TERRESTRIAL  AND  SOLAR  RELATIONS 

conditions  that  injure  the  observations,  it  becomes  necessary  to 
dispense  with  pans  entirely  and  depend  upon  simple  computations. 
There  are  several  methods  of  abbreviation  for  computing  the 
mean  monthly  amounts  of  the  evaporation  from  lakes  and 
reservoirs,  which  make  the  computations  an  insignificant  labor. 
Studies  on  evaporation  from  soil,  sand,  soil  planted  with 
alfalfa,  wheat,  barley,  beans,  have  been  carried  on,  which 
analyze  successfully  the  amount  of  water  lost  under  all  conditions 
throughout  the  year,  from  soils  by  themselves,  and  from  the 
plants  by  themselves.  Thus,  the  transpiration  of  plants  is 
subject  to  accurate  measurement  and  analysis,  and  the  results, 
when  sufficiently  verified,  will  be  of  great  value  to  meteorological 
agriculturalists  and  botanists. 

The  Polarization  of  Sunlight  in  the  Atmosphere 

Common  sunlight  vibrates  indifferently  in  every  plane 
perpendicular  to  its  wave  front,  but  when  it  falls  upon  any 
object,  large  relatively  to  its  wave  length,  a  portion  of  the  light 
is  refracted  and  a  portion  reflected  in  the  plane  of  incidence 
containing  the  incident  and  reflected  rays.  The  vibrations  in 
the  reflected  ray  become  at  least  partially  constrained  to  vibrate 
parallel  to  the  surface  of  reflection,  and  it  is  plane  polarized. 
The  plane  of  the  polarization  is  at  right  angles  to  the  plane  of 
vibration,  and  therefore  contains  the  incident  and  the  reflected 
rays.  If  an  observer  looks  at  any  point  in  the  sky  he  will  receive 
certain  reflected  rays  that  have  proceeded  from  the  sun  to  the 
reflecting  particle  and  to  the  eye,  this  plane  being  the  plane  of 
polarization,  and  the  vibrations  are  at  right  angles  to  it.  If  the 
polarization  is  partial,  and  the  motion  circular,  elliptical,  or  of 
any  other  figure,  components  of  plane  polarized  light  may  be 
constructed  for  this  plane  and  another  at  right  angles  to  it,  so 
that  partially  polarized  plane  vibrations  in  two  directions  at 
right  angles  may  more  or  less  neutralize  each  other  between 
the  limits  0  per  cent  and  100  per  cent.  A  turbid  medium,  such 
as  air  mixed  with  small  solid  particles  of  dust,  ice,  or  even  mole- 
cules, whose  diameters  are  small  relative  to  the  wave  lengths  of 
light,  scatters  and  polarizes  light  by  Rayleigh's  Laws,  in  which 


POLARIZATION   OF   SUNLIGHT  349 

|8  =  the  angle  of  departure  from  the  line  of  incidence  for  the 
reflected  ray. 

(748)  Intensity  of  scattering  =  1  -f-  cos2  /3. 

(749)  Fraction  of  light  polarized  =  — :_     —^. 

Hence  the  maximum  scattering  at  0  =  90°  from  the  sun 
is  twice  as  much  as  in  the  direction  of  the  sun,  /3  =  0;  the 
amount  of  polarization  is  100  per  cent  at  (3  =  90°  from  the 
sun,  and  it  is  0  per  cent  at  /3  =  0°  in  the  direction  of  the  sun. 
In  the  atmosphere  with  the  sun  on  the  horizon,  as  at  the  equinox 
in  the  east,  the  maximum  polarization  is  in  the  zenith,  and 
in  the  vertical  plane  passing  through  the  zenith  and  the  north 
and  south  points.  If  the  solar  point  is  east,  the  antisolar 
point  is  west;  as  the  sun  rises  the  antisolar  point  sinks  below 
the  horizon;  as  the  sun  moves  to  any  other  usual  point  the 
plane  of  polarization  is  that  which  includes  the  sun,  the  point 
of  reflection  in  the  sky,  and  the  eye  of  the  observer,  the  vibrations 
being  generally  at  right  angles  to  this  plane. 

Besides  the  primary  scattering  and  polarization  on  the 
small  particles  in  a  turbid  atmosphere,  it  is  found  that  the  light 
is  only  partially  polarized,  so  that  a  secondary  polarization 
exists  at  right  angles  to  the  primary,  primary  and  secondary 
vibrations,  and  polarizations  at  right  angles  to  each  other,  thus 
tending  more  or  less  to  complete  neutralization  of  plane  polarized 
light  as  the  primary  and  secondary  components  approach 
equality.  There  are  several  such  points  of  neutralization: 
Babinet's  neutral  point  about  15°  to  25°  above  the  solar  point, 
Brews ter's  neutral  point  about  the  same  distance  below  as  the 
sun  rises  above  the  horizon,  and  Arago's  neutral  point  about 
15°  to  25°  above  the  antisolar  point  when  the  sun  is  on  the 
horizon.  The  positions  of  these  points  vary  with  the  position 
of  the  sun  in  the  heavens,  and  the  relative  turbidity  of  the  at- 
mosphere. Since  the  dust  particles  accumulate  chiefly  in  the 
lower  atmosphere,  in  a  stratum  less  than  two  miles  thick,  there 
is  an  apparent  ring  of  special  turbidity  close  to  the  horizon, 
which  causes  the  light  to  be  horizontally  polarized  within  a 


350  TERRESTRIAL  AND   SOLAR  RELATIONS 

few  degrees  of  the  horizon.  Generally,  polarization  is  a  maxi- 
mum in  the  zenith,  and  diminishes  to  the  north  and  south 
horizon  points,  and  from  these  to  the  east  point,  for  a  sun  in 
the  east  and  on  the  horizon.  There  are  numerous  variations 
of  these  principal  results,  due  to  change  in  the  intensity  of  solar 
light  from  radiation,  and  change  in  the  contents  of  turbidity 
in  the  atmosphere. 

The  subject  of  polarization  is  discussed  fully  in  "Tatsachen 
und  Theorien  der  atmospharischen  Polarisation,"  Friedr.  Busch 
and  Chr.  Jensen,  1911.  The  literature  of  the  observations  and 
discussions  is  very  extensive  during  one  hundred  years. 

The  observations  are  made  by  a  polarimeter,  consisting  of 
a  grating  of  parallel  bars  and  spaces,  from  which  the  light  falls 
upon  a  Rochon  prism  which  separates  it  into  the  ordinary  and 
extraordinary  rays.  These  fall  upon  a  Nichol,  and  by  its 
rotation,  there  is  extinction,  or  flattening  of  the  appearance 
of  the  field,  at  four  angles  of  observation.  Thus  for  four  angles 
of  observed  extinction,  the  computation  is  of  the  following  form. 

Cordoba.  (1)  (2)  (3)  (4)  (2-1)       (4-3)       Mean          P% 

Feb.  8,  1912,     11°.3    94°.0     190°.0    278°.0    82°.7     88°.0    85°.4    73.4% 

A  convenient  table  is  prepared  for  obtaining  the  mean  percent- 
age of  polarization  P%  from  the  mean  angles  (2-1)  and  (4-3), 
where  (1),  (2),  (3),  (4)  are  the  successive  readings. 
Various  relations  have  been  traced  out,  such  as: 

(1)  Movement  of  the  altitude  of  the  neutral  points  with  the 
frequency  of  the  number  of  the  sun-spots. 

(2)  Minimum  polarization  at  the  time  of  maximum  tempera- 
ture and  maximum  convection. 

(3)  Maximum  polarization  in  winter  rather  than  summer. 

(4)  Water  drops  have  little  effect  on  the  polarization  at  90° 
from  the  sun;   ice  crystals  and  large  particles  in  that  direction 
decrease  the  polarization  and  increase  the  natural  scattered 
light;  light  that  is  reflected  from  the  earth's  surface,  or  from 
snow  areas,  diminishes  the  polarized  and  increases  the  reflected 
common  light. 

The  relations  between  these  several  terms  have  numerous 
interesting  optical  considerations,  and  they  serve  to  measure 


POLARIZATION   OF   SUNLIGHT 


351 


to  some  extent  the  state  of  turbidity  in  the  atmosphere,  and 
hence  have  value  in  connection  with  the  absorption  and  radiation 
of  solar  energy  in  the  atmosphere. 

The  polarization  at  Daroca,  Spain,  August  19,  26,  1905,  was 
relatively  high  following  rains,  but  it  often  fell  to  40  per  cent  or 


Date 


10       11     Noon     1 


August  19 


August  26 


August  30 


lipse 

Aim 


Sept.21-Oct.10 
U.S.S.  Caesar 


FIG.  71.     Percentage  of  polarization  of  the  sky  light  at  Daroca,  Spain, 
during  August,  and  on  the  U.  S.  S.  Casar,  Sept.  21-Oct.  10,  1905. 

50  per  cent  on  account  of  fine  dust  in  the  air;  that  on  August  30 
shows  the  effect  of  the  passage  of  the  shadow  of  the  solar  eclipse, 
the  sky  having  been  thoroughly  cleaned  of  dust  by  a  rain  on 
August  29;  the  observations  on  the  U.  S.  S.  Casar  during  the 
voyage  from  Gibraltar  to  Norfolk,  frequently  in  the  clear  spaces 
between  cumulus  clouds,  showed  a  normal  high  percentage  of 
polarized  light  at  90°  from  the  sun.  Impurities  from  solid 
particles  produce  natural  light  by  reflection,  fine  particles  and 
gas  molecules  produce  polarized  light. 


352  TERRESTRIAL  AND  SOLAR  RELATIONS 


Solar  Physics 

It  will  be  possible  merely  to  summarize  a  few  important 
points  in  the  subject  of  solar  physics,  in  this  connection,  because 
it  is  very  extensive  in  amount,  and  in  consequence  of  the  fact 
that  much  of  the  theory  is  in  a  conjectural  stage  of  development 
and  is  still  indecisive. 

(1)  It  is  evident  that  the  thermodynamic  equations  employed 
in  the  discussion  of  the  earth's  atmosphere  are  applicable  to 
the  sun's  atmosphere,  by  changing  the  data  in  a  proper  manner. 
Thus,  gravity  becomes  G  =  gQ  X  28.028;    the  pressure  on  the 
photosphere  is  about  five  atmospheres,  so  that  P0  =  5  X  G  pm  En] 
the  temperature  at  the  photosphere  is  apparently  7500°  C.; 
at  the  top  of  the  chromosphere  or  lower  layers  of  the  inner 
corona,  10"  arc  =  7260000  meters,  6900°:  and  top  of  the  inner 
corona,  35"  arc  =  25410000  meters,  6500°.     From  these  data 
for  a  hydrogen  atmosphere,  or  a  calcium  atmosphere,  the  various 
thermodynamic  terms  can  be  computed  as  far  as  the  united 
terms  of  the  velocities'  and  radiations'  energies. 

(2)  The  probable  velocities  in  the  sun-spots,  assuming  that 
they  are  the  stream  lines  of  the  funnel-shaped  or  the  dumbbell- 
shaped  vortices  on  the  upper  plane  of  reference,  on  a  level  with 
the  layer  of  the  photosphere,  can  be  computed  from  the  general 
dimensions  of  the  penumbra  and  umbra,  and  checked  to  some 
extent  by  the  spectroscopic  observations  on  velocities.     The 
vertical  and  the  horizontal  velocities  in  different  layers  of  the 
sun's  atmosphere  are  being  studied  with  the  prospect  of  ultimate 
success  in  a  few  years.     It  may  be  hoped  that  the  radiation 
output  from  the  sun,  computed  from  such  data,  may  be  found  to 
conform  to  the  radiation  energy  at  the  earth  as  derived  from  the 
pyrheliometer  and  the  bolometer  records,  but  much  research 
will  be  required  to  accomplish  this  result. 

(3)  The  rotational  velocity  of  the  sun's  atmosphere  in  different 
latitudes  on  the  level  of  the  photosphere,  and  in  other  higher 
layers  as  already  determined,  indicate  a  very  complex  kind  of 
circulation,  of  an  entirely  different  type  from  that  in  the  earth's 


SOLAR  PHYSICS  353 

atmosphere.  The  latter  consists  of  a  thin  shell  heated  on  the 
tropics,  and  acquiring  an  approximately  steady  type  of  equili- 
brium, as  heretofore  explained,  while  the  sun  has  maximum 
velocity  at  the  equator  diminishing  to  the  poles  on  the  level  of 
the  photosphere,  and  increasing  upward  in  all  latitudes.  Since 
the  integral  in  every  small  column  along  a  radius  extended  must 
conform  to  the  gravity  integral,  which  is  the  sum  of  the  pressure, 
circulation  across  it,  and  radiation  through  it,  there  is  an  op- 
portunity to  determine  these  terms  through  an  approximation 
by  trials. 

(4)  Table  80  contains  a  convenient  series  of  transformations 
between  sidereal  and  synodic  periods.     Table  81   contains  a 
collection  of  the  observed  synodic  periods  of  rotation  in  different 
latitudes.     Bigelow's  data  from  the  prominences  refer    to  the 
higher  levels  of  the  sun's  atmosphere,  because  they  are  seen 
projected  above  the  chromosphere.     The  acceleration  in  the 
polar  region  over  the  velocities  devised    from  spectrum  dis- 
placement lines  is  probably  correct,  because  the  spectrum  lines 
are  all  located  at  lower  levels.     Some  of  the  Mt.  Wilson  data 
are  in  conformity  with  this  result. 

(5)  The  magnetic  data  at  the  earth,  as  already  indicated, 
produce  a  synodic  period  of  26.68  days  at  the  sun's  equator, 
conforming  closely  to  the  general  mean  value  872'  or  26.58  days 
from  the  eight  researches  quoted.     The  Zeeman  effect  has  been 
detected  by  Professor  Hale  in  the  sun-spots,  due  to  the  rotation 
of  electric  ions  in  the  tube  of  the  vortex.     This  proves  that 
electric  ions  in  circulation  produce  magnetic  field  at  solar  tem- 
peratures.    Hence,   the  interior  of  the  sun,   if  polarized  into 
rotation  filaments  by  its  circulation,  by  rotation  on  its  axis,  and 
processes  of  radiation,  is  probably  magnetized  throughout  its 
mass,  in  much  the  same  way  that  the  earth  carries  an  internal 
and  external  magnetic  field,  though  its  interior  is  at  a  high  tem- 
perature.    There  is  evidence  of  such  spherical  magnetism  in  the 
shapes  of  the  polar  rays  seen  in  the  minimum  activity  of  the 
coronal  formation,  where  the  observed  rays  from  the  sun,  seen 
in  projection,  conform  to  the  lines  of  force  surrounding  a  sphere, 
supposing  that  they  are  generated  chiefly  in  a  polar  ring  about 


354 


TERRESTRIAL  AND    SOLAR  RELATIONS 


TABLE  80 

THE  SIDEREAL  AND  SYNODIC  PERIODS  OF  THE  ROTATION  OF  THE  SUN  FOR 
CERTAIN  ASSUMED  DAILY  ANGULAR  VELOCITIES.    -=  —  —  =  -$=  k  —n. 

1  £L          O 


Daily 
Angular 
Velocity 

X 

Daily 
Angular 
Velocity 
in  Degrees 

f 

Sidereal 
Period 
in  Days 

r 

Angular 
Velocity  of 
the  Sun 
in  Days 

h* 

Angular 
Velocity  of 
the  Earth 
in  Days 
1 
~=n 

Synodic 
Velocity  of 
the  Sun 
in  Days 

j  =  k-n 

Synodic 
Period 
in  Days 

S 

1 

700 

o 

11.67 

30.86 

.03241 

.00274 

.02967 

33.70 

705 

11.75 

30.64 

.03264 

" 

.02990 

33.44 

710 

11.83 

30.42 

.03287 

" 

.03013 

33.19 

715 

11.92 

30.21 

.03310 

" 

.03036 

32.94 

720 

12.00 

30.00 

.03333 

" 

.03059 

32.69 

725 

12.08 

29.79 

.03357 

" 

.03083 

32.44 

730 

12.17 

29.59 

.03380 

" 

.03106 

32.20 

735 

12.25 

29.39 

.03403 

" 

.03129 

31.96 

740 

12.33 

29.19 

.03426 

" 

.03152 

31.73 

745 

12.42 

28.99 

.03449 

n 

.03175 

31.50 

750 

12.50 

28.80 

.03472 

" 

.03198 

31.28 

755 

12.58 

28.61 

.03495 

" 

.03221 

31.05 

760 

12.67 

28.42 

.03518 

11 

.03244 

30.83 

765 

12.75 

28.24 

.03542 

" 

.03268 

30.61 

770 

12.83 

28.05 

.03565 

" 

.03291 

30.39 

775 

12.92 

27.87 

.03588 

" 

.03314 

30.18 

780 

13.00 

27.69 

.03611 

" 

.03337 

29.97 

785 

13.08 

27.52 

.03634 

" 

.03360 

29.76 

790 

13.17 

27.34 

.03657 

n 

.03383 

29.56 

795 

13.25 

27.17 

.03681 

11 

.03307 

29.35 

800 

13.33 

27.00 

.03704 

11 

.03430 

29.15 

805 

13.42 

26.83 

.03727 

" 

.03453 

28.96 

810 

13.50 

26.67 

.03750 

" 

.03476 

28.77 

815 

13.58 

26.50 

.03773 

11 

.03499 

28.58 

820 

13.67 

26.34 

.03796 

11 

.03522 

28.39 

825 

13.75 

26.18 

.03819 

" 

.03545 

28.20 

830 

13.83 

26.02 

.03843 

" 

.03569 

28.01 

835 

13.92 

25.87 

.03867 

" 

.03592 

27.83 

840 

14.00 

25.71 

.03889 

"- 

.03615 

27.66 

845 

14.08 

25.56 

.03912 

" 

.03638 

27.49 

850 

14.17 

25.41 

.03935 

11 

.03661 

27.32* 

855 

14.25 

25.26 

.03958 

11 

.03684 

27.14 

860 

14.33 

25.12 

.03982 

" 

.03708 

26.97 

865 

14.42 

24.97 

.04005 

" 

.03731 

26.80 

870 

14.50 

24.83 

.04028 

" 

.03754 

26.64f 

875 

14.58 

24.69 

.04051 

" 

.03777 

26.48 

880 

14.67 

24.55 

.04074 

it 

.03800 

26.32 

885 

14.7.5 

24  .41 

.04097 

" 

.03823 

26.16 

890 

14.83 

24.27 

.04120 

" 

.03846 

26.00 

895 

14.92 

24.13 

.04144 

it 

.03870 

25.84J 

900 

15.00 

24.00 

.04167 

11 

.03893 

25.69 

930 

15.50 

23.22 

.04306 

n 

.04032 

24.80§ 

*Lat.  .13°,  spots.          f  Equator  of  sun.          $  Terrestrial.          §  Hydrogen. 


SOLAR   PHYSICS 


355 


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356  TERRESTRIAL  AND  SOLAR  RELATIONS 

23  degrees  distant  from  the  coronal  pole.  This  system  has 
been  traced  from  one  epoch  to  another,  through  several  eclipses 
from  1878  to  1905  at  least,  as  if  the  synodic  period  26.68  days 
was  also  fundamental  in  producing  the  aspect  they  present  upon 
the  series  of  eclipse  photographs.  A  model  was  constructed 
of  such  a  magnetic  field,  and  turned  by  its  astronomical  co- 
ordinates into  the  required  positions  on  the  days  of  the  several 
eclipses.  The  coincidence  in  position  between  the  pole  of  the 
sun,  pole  of  the  earth,  pole  of  the  corona,  and  its  stream  lines  as 
parts  of  a  spherical  magnetic  field,  are  too  striking  to  be  over- 
looked. Superposed  upon  this  deep-seated  magnetic  field,  em- 
bracing the  entire  interior  of  the  sun,  is  a  strong  electrostatic 
surface  field  with  its  rays  in  normal  directions,  and  in  many  dis- 
torted positions.  These  two  fields  interplay  among  the  forces  of 
circulation  and  radiation  to  produce  the  numerous  fantastic  forms 
seen  on  the  edge  of  the  sun.  Astron.  Soc.  Pac.  No.  27,  1891. 

The  Spherical  Astronomy  of  the  Sun 

It  is  necessary  to  give  a  brief  account  of  the  variable  relations 
due  to  the  rotation  of  the  sun  on  its  axis,  and  the  revolution  of 
the  earth  in.  its  orbit  about  the  sun.  If  the  spherical  conditions 
of  Fig.  72  be  transferred  to  a  small  rubber  ball,  it  will  greatly 
facilitate  the  study  of  this  complicated  branch  of  solar  physics. 
The  photographs  of  the  sun  give  pictures  which  must  be  inter- 
preted in  terms  of  spherical  co-ordinates,  and  this  is  a  great 
labor  of  computation,  where  any  large  number  of  points  are  to 
be  considered.  Some  mechanical  devices  have  been  used  for 
securing  heliocentric  co-ordinates  approximately,  but  for  defini- 
tive work  the  micrometer  measurements  must  be  employed  with 
accuracy.  As  the  earth  passes  around  the  sun  the  aspect  of  the 
disk  undergoes  an  annual  periodic  change  which  must  be  followed, 
and  as  the  sun  rotates  on  its  axis  the  positions  of  the  spots, 
faculaa,  and  prominences  change  from  day  to  day.  The  following 
definitions  and  formulae  can  be  very  readily  verified  from  the 
diagram,  and  by  Chauvenet's  treatise  on  Spherical  Trigonometry. 
The  angle  H  is  the  apparent  projection  of  SC  K  on  a  plane 


SPHERICAL  ASTRONOMY   OF   THE    SUN 


357 


perpendicular  to  the  plane  of  sight,  the  angle  G  is  the  apparent 
projection  of  K  C  E  on  the  same  plane,  the  angle  P  is  the  position 
angle  of  the  spot  2  from  the  north.  E,  counted  positive  eastward, 
so  that  X  =  H  +  G  -f  P,  the  position  angle  from  the  sun's  pole 
S.  At  the  same  time  p  is  the  angular  distance  of  S  from  C,  as 
measured  at  the  center  of  the  sun.  The  prime  meridian  is  the 
central  meridian  at  one  adopted  epoch,  mean  midnight,  Decem- 
ber 31,  1853  (Carrington) ;  Greenwich  mean  noon,  June,  13.72 
(1887),  Bigelow.  The  rotation  periods  of  the  sun  change  in 
latitude,  from  a  maximum,  26.68  days  synodic  at  the  equator, 
to  a  minimum,  about  30.00  days  at  the  poles.  Carrington's 
adopted  period  of  rotation  is  applicable  to  latitudes  =  12°,  and 


O=Center  of  Sun 
C  =  Center  of  Disk 
2=  Position  of  Spot 


''  (l.d.)  Sun's  Equator 


Ecliptic 

(cC.5.)  Earth's  Equator 
L  from  N  to  C—  >-  Center 
I    from  N  to  "2  -*•  Spot 
/'  from  N  to  M->  Prime  Meridian 


FIG.  72.    Spherical  positions  on  the  sun's  equator. 

represents  the  average  period  of  rotation  for  the  sun-spots  in 
that  latitude  alone.  The  radiation  effects  measured  at  the  earth 
are  for  the  equatorial  period. 

Poles.  5  =  sun's    equator;    E  =  earth's  equator;    K  = 

Ecliptic. 

Inclinations.    7  =  sun's  equator  to  the  ecliptic,  7°.  15'  =  KO'S. 
co  =  earth's    equator    to  the   ecliptic,  23°.27'  = 

EO'  K. 


358  TERRESTRIAL  AND   SOLAR  RELATIONS 

H  +  G  =  earth's  equator  to  the  sun's  equator,  26°  20'  = 

EO'S. 
H  =  the  projection  of  S  0'  K  at  the  center  of  the 

disk  C. 
G  =  the  projection  of  K  O'  E  at  the  center  of  the 

disk  C. 

P  =  the  position  angle  of  2  =  EC  2. 
X  =  H  +  G  +  P. 

The  positive  direction  of  the  angles  H,  G,  P}  X  is  through  the 
east. 

p  =  the  heliocentric  angle  of  2  from  C  =  C  0'  2  . 


Co-ordinates  of  the  center  of  the  disk  C. 

at 


L  =  longitude  from  the  node  N  on  the  sun's  equator, 

NO'L- 


D  =  latitude  from  the  sun's  equator  =  C.  L. 

f  E  =  right  ascension  from  the  node  ^  on  the  earth's 

equator,  TO'  E. 
[  F  =  declination  from  the  earth's  equator  =  C  E. 

f  O  =  sun's  celestial  longitude  from  the  node  T  on  the 
Ecliptic]  ecliptic,  T  Of  C. 

(^  ...  =  the  latitude  is  zero  =  C  C. 

Co-ordinates  of  the  spot  or  point  2  on  the  disk. 

c     ,  /  =  longitude  from  the  node  N  on  the  sun's  equator, 

NO'L 
equator       .      ,    .     ,    „ 

^  a  ==  latitude  from  the  sun's  equator  =  /  S. 

.  ,    (  a  =  right  ascension  from  the  node  T  on  the  earth's 
Earth's  ^  ^ 

-j  equator  T  0  a. 

[   8  =  declination  from  the  earth's  equator,  a  S. 

A  =  heliocentric  longitude  from  the  node  T  on   the 
Ecliptic  -j  ecliptic,  TOrA. 

/?  =  heliocentric  latitude  from  the  ecliptic  =  A  2. 
/'=  longitude  of  the  prime  meridian  from  N  on  sun's 
equator. 


SPHERICAL  ASTRONOMY  OF  THE   SUN  359 

L-l  =  heliocentric  longitude  from  the  central  meridian, 
or  difference  of  heliographic  longitudes  of 
center  of  disk  and  spot. 

L-V  —  heliographic  longitude  of  center  of  disk  from  the 
prime  meridian. 

/-/'  =  heliographic    longitude    of    2    from    the    prime 

meridian. 
Reduction  of  the  photographic  plate. 

R  =  solar  radius  on  the  photograph. 
R'  =  solar  radius  corrected  for  distortion  on  the  plate. 
R"  =  solar  radius  as  given  in  the  ephemeris. 
r  =  the  perpendicular  distance  of  S  from  the  line 

COO',  =  02. 

rf  =  the  measured  distance  of  the  spot  from  the  cen- 
ter of  the  sun-picture,  corrected  for  distortion. 
p'  =  the  angular  distance  0  2  as  seen  from  the  earth. 

(750)  p'  =  jr  R".  ~r  =  sin  0'  2  2'  =  sin  (p  +  p'). 

(751)  p  +  P'=  sin"1  —-.  p  =  sin'1  ~-  -  p'. 

From   the  right  spherical  triangle  TEC  (Chauvenet,  p. 
172,  88), 

(752)  tan  G  =  tan  co  cos  O ;  from  cos  O  =  cot  co  cot  (90-£). 
From  the  right  spherical  triangle  N  L  C  (Chauvenet,  p. 

172,  88), 

(753)  tan  H  =  cos  (O  -  AQ  tan  7;  from  cos  (O  -  N)  =  cot  I 
cot  (90°-#). 

From  the  right  spherical  triangle  N  L  C  (Chauvenet,  p. 
171,  86), 

(754)  sin  D  =  sin  (O  -  N)  sin  7. 

From  the  right  spherical  triangle  N  L  C  (Chauvenet,  p. 
171,  87), 

(755)  tan  L  =  tan  (O  -  N)  cos  7. 

From  the  spherical  triangle  S  C  2,  two  sides  and  the  included 
angle  known,  90-Z),  X,  p  (Chauvenet,  p.  179,  M), 


360  TERRESTRIAL  AND   SOLAR  RELATIONS 

(756)  sin  d  =  cos  p  sin  D  +  sin  p  cos  DcosX. 

From  the  spherical  triangle  S  C  2,  from  the  two  sides  and 
angle  opposite  one  of  them  (Chauvenet,  p.  193,  148), 

(757)  Sin  (L  -  /)  =  sin  X  sin  p  sec  d. 

From  the  right  spherical  triangle  T  E  C  (Chauvenet,  p.  171, 
86,  87), 

(758)  sin  F  =  sin  O  sin  co. 

(759)  tanE  =  tan  O  cos  co. 

T  =  the  fraction  of  a    revolution    executed  by  the 
prime  meridian  at  a  given  date. 

/  =  time  from  the  epoch,  June  13.72,  1887. 

K  =  the  mean  angular  velocity  of  the  sun  on  its  axis. 

n  =  the  mean  angular  velocity  of  the  earth  around 

the  sun. 
K  —  n  =  the  synodic  angular  velocity  of  the  sun. 

m  =  the  complete  number  of  sidereal  rotations  of  the 
prime  meridian  M  since  the  epoch. 

(760)  T  =  -  m  (Bigelow): 


(761)          l'  =  TX  =  T  X  14°.4783  (Bigelow). 


To  transform  (A  .  /?)  to  (L  .  D),  compute  the  auxiliary  angle  a, 

(762)  tan  a  =  sin  A  cot  0.     Then, 

/«    \  si*1  (I  +  0) 

(763)  tan  L  =  -  ^  --  •  tan  A. 

sin  a 

(764)  sinZ?  = 


cos  a 


The  Magnetic  Fields  of  the  Earth  and  the  Sun 

The  sun  presents  many  aspects  of  a  magnetized  sphere,  with 
the  positive  pole  on  the  south  side  of  the  sun's  equator;  the  earth 
is  a  magnetized  sphere  with  its  positive  pole  on  the  south  side 
of  the  equator.  An  isolated  (supposed)  positive  (+)  magnetic 
pole  tends  from  the  south  side  to  the  north  side  of  the  sun's 


SPHERICAL  ASTRONOMY   OF   THE    SUN 


361 


equator,  and  from  the  south  side  to  the  north  side  of  the  earth's 
equator. 


(x)     Magnetic  pole  near  the  south  geographical  pole. 
FIG.  73.    Magnetic  coordinates  and  component  forces. 

The  following  formulas  are  convenient  for  reference: 

(765)  Mass  =  M  =  -  x  R*. 

o 

(766)  Mass  Potential  P  =  y  =  -g  TT  Rs-  — 

dP       4      rR* 

(767)  Vector  Potential  Ve  =  -  I  -^  =  -g  *  I  -^  *  = 


—  TT  /  —  cos  6  (external). 


(768) 


(internal), 


Exterior  Forces 


e  , 

(770)    X.,—  --.+      */ 


a* 


4      I?3 

—  TT  -r  /  .  3  sin  e  cos 

O         IT* 


362 

(771) 


TERRESTRIAL  AND   SOLAR  RELATIONS 

dV.  4         R*     Zzy 

+  s"71"7^  '~^~          =  0.since;y  =  0. 


dy  3         r3       r* 


(772)     Z   =  Fn  =       Z0  cos  0  +  X0  sin  0  =  -     TT  7 

o  T" 

(-  cos  0  +  3  cos3  5  +  3  sin2  0  cos  0) 


r 

=  —  x  /  —  •  2  cos  0. 


(773)     X    =  Ft  =  -  Z0  sin  0  +  X0  cos  0  =  -|  TT  7  ^ 

(sin  0-3  cos2  0  sin  0  +  3  sin  0  cos2  0) 


(774) 


Z02  = 


r 
=  —  TT  /  —  r  .  sm  0. 

o  f-5 


=         TT  7  (l  +  3  cos*  0). 


Surface  R  =  r 


(775)     Ze  =  -  TT  7  .  2  cos  0 

o 


(776)    Xe  =  -     T  7  .  sin  0  ' 


Equator 


8      r 

o-  *  /. 


Pole  - 

'e   =  0. 

4 


=  0. 


Interior  Forces 


C778)  ^  - -•  7?  T  * 


(780)    Line  of  Force 


=  constant. 


SPHERICAL  ASTRONOMY   OF  THE   SUN  363 

(781)     Equipotential  Surface  V  =  —K  R3 — <r~  =  constant. 

O  T 

A  magnetized  sphere  may  be  induced  by  two  layers  of 
positive  and  negative  masses,  circulating  surface  electric  currents, 
two  positive  and  negative  poles  placed  near  the  center,  and  by 
other  physical  devices,  as  rotating  Ampere  electric  currents  in 
the  interior  around  lines  parallel  to  the  axis,  in  which  case  the 
sphere  is  polarized.  This  is  probably  the  case  for  the  earth  and 
for  the  sun.  For  surface  exterior  currents,  as  in  the  earth's 
atmosphere,  it  is  evident  that  variation  in  the  east- west  direction 
changes  the  strength  of  the  existing  magnetic  field  along  the  lines 
of  force.  The  vectors  of  Fig.  68,  69,  can  only  be  produced  by 
north  and  south  ionization  currents,  as  there  indicated,  distinct 
from  east  and  west  currents  on  the  normal  field.  The  asymetric 
position  of  the  sun's  magnetic  poles  deduced  by  Bigelow  in 
1891  has  been  confirmed  by  direct  observation  at  the  Mt. 
Wilson  observatory,  1913. 

Conclusion 

The  general  functional  relations  between  the  incoming  solar 
radiation,  the  portion  of  it  transformed  into  ionization  currents, 
the  magnetic  disturbing  vectors  depending  upon  them,  all  re- 
main to  be  discovered.  It  has  been  possible  to  indicate  in  this 
Treatise  some  of  the  important  elements  in  these  problems,  and 
it  is  hoped  that  the  formulas  and  methods  of  discussion  here 
adopted  will  greatly  facilitate  the  pursuit  of  such  researches  by 
many  students.  The  practical  side  of  the  matter  consists  in 
the  development  of  the  branch  of  Meteorology  and  Solar  Physics 
which  will  culminate  in  the  ability  to  predict  the  seasonal  climatic 
conditions  likely  to  prevail  during  the  coming  year  in  the  several 
large  agricultural  regions  of  the  earth.  The  extent  and  scope  of 
these  subjects  are  so  great  that  the  co-operation  of  many  institu- 
tions and  national  offices  will  be  essential  for  the  success  of  so 
important  an  enterprise. 


CHAPTER  VII 

Extension  of  the  Thermodynamic  Computations  to  the  Top  of 

the  Atmosphere 

Remarks  on  the  Bouguer  Formula 

THE  computations  on  the  thermodynamic  data  of  the  earth's 
atmosphere  have  heretofore  in  the  examples  been  limited  to 
about  20,000  meters,  but  it  is  very  desirable  to  extend  them  to 
the  vanishing  plane  of  the  gaseous  envelope.  The  following 
summary  illustrates  a  method  for  accomplishing  this  purpose, 
and  the  results  afford  approximate  material  for  further  important 
researches.  These  concern  the  problems  of  the  solar  radiation 
and  its  absorption,  together  with  the  correlative  terrestrial 
radiation  and  its  absorption.  It  is  especially  proposed  to 
test  the  result  that  the  "solar  constant,"  as  derived  from 

pyrheliometer  observations  is  about  1.92  -^-~- — r-,  or  whether 

cm.2  mm.' 

it  is  about  twice  as  much,  in  conformity  with  the  bolometer 
data.  The  Bouguer  formula  of  reduction,  in  the  preceding 
notation, 


has  been  shown  to  contain  two  unknown  terms  /o,  mo,  which 
together  determine  the  source  of  the  effective  radiation,  so  soon 
as  their  details  are  understood.  It  has  been  customary  to 
identify  /0  with  the  solar  radiation  falling  on  the  outer  limit  of 
the  atmosphere  for  which  it  is  assumed  that  m0  =  i  and  disappears 
from  the  equation,  enabling  it  to  be  solved  for  70.  This  is,  in 
fact,  a  special  interpretation  by  assumption,  and  doubt  is  thrown 
upon  it  by  the  bolometer  ordinates,  since  these  require  an  energy 
spectrum  curve  of  about  T  =  6,900°  (3.88  calories),  while 
/o  =  1.92  calories  is  satisfied  with  5,800°  for  the  effective  solar 
temperature.  It  has,  furthermore,  been  shown  by  the  pyrheliom- 

364 


FIRST  DISTRIBUTION   OF   TEMPERATURE  365 

eter  observations  that  /o  is  not  a  constant  quantity,  that  is,  a 
source  of  radiation  independent  of  atmospheric  conditions  as  to 
aqueous  vapor,  dust  and  ice  contents,  not  now  referring  to  the 
small  variations  of  the  radiation  energy  emitted  at  the  sun  itself. 
The  evidence  is  that  as  the  coefficient  of  transmission  p  dimin- 
ishes in  value  the  term  70  is  progressively  depleted.  The  extra- 
polated points  /o  on  the  axis  of  ordinates  do  not  concentrate  at 
/o  =  1.92  calories.  It  follows  that  /o  is  a  terrestrial  variable, 
and  it  is  the  purpose  of  this  chapter  to  investigate  further  the 
properties  of  the  terms,  70,  mQ. 

I.  FIRST  DISTRIBUTION  OF  TEMPERATURE 

The  balloon  ascension,  Huron,  So.  Dak.,  September  1.  1910 

The  balloon  ascension,  Huron,  So.  Dak.,  September  1,  1910, 
reached  a  height  of  30,000  meters,  recording  temperatures  7", 
but  without  velocities  q.  These  have  been  provisionally  sup- 
plied by  taking  the  mean  values  for  ten  other  ascensions  in  order 
to  balance  approximately  the  terms  of  the  gravity  equation, 

(196)  g  (*  -  *)  -  -  ^-p^  -  J  fo2  -  <?o2)  -  (Qi  -  Co). 

^10 

Table  82  contains  the  adopted  T,  q,  from  which  all  the  other 
terms  were  computed  up  to  30,000  meters,  and  the  results  were 
plotted  on  large  diagrams  of  which  Figs.  74,  75,  are  reduced 
copies.  The  method  of  extending  the  T  data  to  the  higher 
planes  is  as  follows:  On  the  30,000-meter  plane  T  =  232°.3, 
P  =  1352  (B  =  10.14  mm.  as  computed,  while  the  automatic 
register  recorded  B  =  10.70  mm.),  p  =  0.05512,  R  =  105.61, 
the  check  equation  P  =  p  R  T  being  perfectly  satisfied.  At  first 
the  P  and  p  curves  were  extended  graphically,  and  from  these 
values  up  to  40,000  meters  approximate  values  of  T  were  com- 
puted, carrying  the  T  curve  around  the  corner  of  the  isothermal 
region  into  a  very  rapid  gradient.  These  temperatures  are 
necessary  to  continue  the  P,  p,  curvatures  at  that  elevation. 
Then  all  the  other  terms  were  computed  directly  from  T  and  the 
diagrams  were  extended.  At  40,000  meters  T  =  170°.5,  P  = 


366 


EXTENSION  OF  THERMODYNAMIC  COMPUTATIONS 


§fl 


c/3 


a  gs 

M    S3 


t*    g 


*!? 

II 

u 

Is 

OK 
§5 

11 

O  ^ 

I 

M 


to 


Tempe 
ture 


rococococococococococococoro.cocococococococo 


1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1 


CO 

»o  T-H 

i—  i  O 
i-H*O 


i—  ICOCOO5(NCOO5(M>O 


COO5(M>OOOi—  IIO 
I—  li—  (C^tNfNCOCO 


*OI—  i(MOO 
i—  IIOO5OO 


COOOiOOi<MOi—  IT—  iiOrH 


.T 


OOOOO<NCOOO»O 


iOi—  lOSGOOOOiOi—  i<N^f>O 
CO  CO  C^l  C^  C^  Ol  CO  CO  CO  CO  CO 


^ 

vsx^f'OiO'—  !COOiCOTtiCO 
y\/N    c>q-^iu7if>.cOcOOi»Ot^ 


O51>«iOCOCO(NOO 
COO'-HCOcO'—  i  i—  I 


rH  <N 


X   X  ^ 

GO  00  i—  1  CO  l>«  O5  CO 

rH    TH   t^   10   TJH   CO   CO   <M 


§ 


FIRST  DISTRIBUTION   OF   TEMPERATURE  367 


OO   00   ^2   *—  4    C^    CO 


cococococococococococococococococooocococococococococococccococo 


I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I 


I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I 


CO 


<N 


OiCOOO5C<l 


^ 

CO 


co  <N  C5  as  0   t    Tt<  <M  i-  o  o  0   0   o  0   o  <o  co  o  t    •*  o  o 

C^COCOC^C^C^C<lC^C<IC^C<JrHrHrHrHC^C^CO^TjHUOO^ 


368 


EXTENSION  OF   THERMODYNAMIC   COMPUTATIONS 


279  (B  =  0.002095  mm.),  P  =  0.01796,  R  =  91.20,  and  P  = 
p  R  T  is  again  exactly  satisfied.  These  data  indicate  that  the 
vanishing  plane  of  the  atmosphere  is  approaching,  and  a  trial  was 
made  of  z  =  50,000  meters  for  T  =  0°  the  approximate  temper- 
ature of  space.  Further  experience  with  the  data  indicates  that 


50000 


40000 


FIG.  74.     The  dynamic  data  to  the  top  of  the  atmosphere  at  50,000  meters, 
from  the  balloon  ascension,  Huron,  S.  D.,  Sept.  I,  1910 


P  =  Pressure  in  kilograms  per  square  meter. 
—  K  =  Radiation  energy. 

p  =  Density  in  kilograms  per  cubic  meter. 
T= Temperature  in  absolute  degrees. 

rj=  Temperature  before  the  formation  of  the  isothermal  region. 
Cp  =  Specific  heat  ratio  at  constant  pressure. 

The  balloon  ascension,  Huron,  S.  D.,  September  I,  1910,  reached  the 
elevation  30,000  meters.  The  lines  for  P,  —  K,  />,  Cp,  and  T  (up  to  the  iso- 
thermal level)  converge  at  a  point  on  the  vanishing  plane  50,000  meters.  By 
,a  series  of  trials  T  was  determined  such  as  to  make  the  P,  —  K,  p  curves  run 
out  smoothly  to  a  common  vanishing  point. 


FIRST  DISTRIBUTION   OF   TEMPERATURE 


369 


the  vanishing  plane  is  nearer  90,000  meters,  but  the  example  is 
reproduced  because  several  interesting  points  are  illustrated. 
The  temperature  T  at  any  level,  whether  observed  or  adopted, 
can  always  be  fully  checked  by  the  combination  of  these  two 
equations.  If  there  is  an  important  residual  for  g  (zi  —  z0)  it 
must  be  due  to  the  fact  that  an  erroneous  T  was  ascribed  to  the 
level.  In  this  case  the  residuals  are  large — from  41,000  to  50,000 
meters,  but  this  can  be  remedied  by  further  trial  computations  in 


50000 


40000 


30000 


20000 


10000 


000 


(QrQul/v*' 


(Oi-O 


(u,- 


Uo) 


(W:-W0) 
(Ui-Uo) 


2000.       4000.       6000.       8000.       10000.      12000. 

3.50   3.&0   3.70  3.80 


FIG.  75.  The  thermodynamic  and  radiation  data  to  the  top  of  the  atmos- 
phere at  50,000  meters,  from  the  balloon  ascension,  Huron,  S.  D.,  Sept. 
I,  1910. 

(Qi  —  Qo)  =The  free  heat  transmitted  by  the  solar  radiation. 
(Wi—Wo)  =The  external  work  of  expansion. 
(Ui—  Uo)  =The  inner  energy  of  molecular  motion. 

a=The  exponent  in  the  radiation  equation. 
(Ei— Eo)  =The  total  incoming  free  heat. 
(A  i— Ao)  =The  absorbed  portion  of  it. 

(Qi-  <2o)  =  (W,  -  W,}  +  ( Ui  -  C/o) 


370  EXTENSION  OF   THERMODYNAMIC   COMPUTATIONS 

connection  with  other  high-level  balloon  ascensions.  It  should 
be  noted  in  this  example  that  the  value  of  R  =  268.46  at 
46,000  meters  has  resumed  nearly  its  surface  value,  287.03.  This 
means  that  the  thermodynamic  system  has  been  subjected  to 
great  heating  influences  in  the  levels  above  40,000  meters,  and 
this  is  easily  seen  by  studying  the  data  of  Table  82,  and  Figs. 
74,  75.  The  P,  —  K,  p,  curves  run  out  smoothly  to  the  vanishing 

plane,  but  the  Cp  =  R  ~^  (Qi  -  ft),  (Wi  ~  W0),  (U,-  Z70), 

log  c,  a,  curves  all  are  much  distorted  in  the  strata  above  40,000 
meters.  This  is  the  thermodynamic  result  of  heating  the  rarefied 
gases  of  the  outer  levels  from  T  =  0°  on  the  vanishing  plane 
to  T  =  208°.2  at  38,000  meters.  The  so-called  isothermal 
layer  from  11,000  to  38,000  meters  is  merely  that  part  of  the 
atmosphere  where  the  surviving  solar  radiation  nearly  balances 
the  terrestrial  radiation.  Above  38,000  meters  is  a  region  of 
very  powerful  absorption  of  the  solar  radiation,  in  which  the 
energy  of  the  short  waves  of  the  solar  spectrum,  *  =  0.00  /* 
to  0.40  M,  are  absorbed,  the  energy  being  used  in  heating  this 
stratum  through  about  208  degrees.  This  is  the  true  albedo 
of  the  earth's  atmosphere,  and  it  amounts  to  nearly  one-half  of 
the  total  solar  radiation  falling  on  the  earth's  outermost  stratum 
of  atmosphere.  If  a  line  be  drawn,  T1,  from  the  lower  part  to 
the  upper  part  of  the  T-curve,  the  enclosed  area  between  T  and 
T1  represents  the  region  of  absorption,  and  the  isothermal  curve 
to  the  right  of  it  is  one  of  its  three  boundaries.  This  area  can 
be  studied  for  several  thermodynamic  conditions. 

The  divergence  between  the  pressure  curve  P  and  the 
radiation  energy  curve  —  K  is  due  to  the  free  heat  per  volume 
«?i  ~  Co)  -  (Wi  -  TFo)  +  (tfi  -  Z7o),  where  (ft  -  ft)  repre- 
sents the  excess  of  the  inner  energy  (Ui  —  U0)  over  the  work  of 
expansion  (Wi  —  WQ)  .  At  the  points  where  the  P  and  —  K  curves 
coincide,  as  at  the  surface,  there  is  no  free  heat  at  liberty  to  move. 
In  the  case  of  an  adiabatic  atmosphere,  (ft  —  ft)  =  0,  and  there 
is  no  radiation.  It  should  be  carefully  noted  that  meteorological 
discussions  and  the  tables  in  common  use  have  been  heretofore 
uniformly  based  upon  strictly  adiabatic  formulas.  Since  the 


FIRST  DISTRIBUTION   OF   TEMPERATURE  371 

adiabatic  and  non-adiabatic  systems  are  identical  at  the  surface, 
this  distinction  is  not  important  there,  but  because  these  systems 
diverge  to  a  large  amount  in  the  upper  atmosphere  the  inference 
follows  that  all  meteorological  tables  based  upon  R  =  constant 
produce  erroneous  values  of  the  density  above  the  surface.  For 
example,  with  R  =  constant,  g  (zi—  ZQ)  =  —  Cpa  (Ta—  T0)  = 

— ,  and  from  this  comes  the  adiabatic  gradient  77—  = 

PaO  Cpa 

-    (a  ~  T^  =  -  9.870  per   1,000  meters.     If,   on   the  other 

(21  -  z0) 

hand,  the  terms  for  circulation  —  \  (qi2  —  q<?)  and  for  radia- 
tion —  (Qi  —  QQ)  are  to  be  introduced,  this  can  only  be  done  by 
changing  pao  to  pio,  since  the  pressure  term  (Pi  —  PQ)  is  fixed 
by  direct  measures  at  heights  which  do  not  correspond  with  any 
adiabatic  temperature  gradient.  It  follows  that  we  have 

another  value  of  the  specific  heat  ratio  Cpio  which  differs  from 

•p       -p 

Cp^   so   that  —  Cpio    (Ta  —  TO)  =  —   •  -  and    the   non- 

|°10 

adiabatic  pio  differs  from  the  adiabatic  pa.  By  subtraction  we 
find  that, 

(198)    -  (Cpa  -  CplQ)  (Ta  -  T0)  =    -\  (qi2  -  qo2)  -  (Qi  -  Qo), 

as  heretofore  indicated.  If  for  the  moment  we  assume  that 
~"  i  (<?i2  ~~  #o2)  =  0,  as  is  the  case  at  the  top  and  at  the  bottom 
of  the  atmosphere,  there  are  two  extreme  values  of  (Qi  —  Q0) : 

1 .  At  the  vanishing  plane,  for  Cpio  —  0, 

-  (Qi  ~  <2o)  =  -  (Cpa  -  Cpio)     (Ta  -  To)  =  993.58  X  9.87 
=  9806,  per  1,000  m. 

2.  At  the  surface,  for  Cpw  =  Cpa, 

~  (Ql  -  <2o)    =    -   (Cpa  ~  Cpio)   (Ta  -  T0)   =  0. 

The  adiabatic  formula  requires  that  Cpio  =  Cpa  from  the 
surface  to  the  vanishing  plane,  which  is  in  disagreement  with 
the  observed  temperature  gradients. 


372 


EXTENSION   OF   THERMODYNAMIC   COMPUTATIONS 


Illustrations  of  the  Use  of  Erroneous  Densities 

In  order  to  illustrate   the  amount  of  divergence  between 
pa  and  p,  we  have  for  this  balloon  ascension — 

TABLE  83 

THE  DIFFERENCE  BETWEEN  />  (NON-ADIABATIC)  AND  pa  (ADIABATIC) 

DENSITY 


z 

ft 

Pa 

50000 

0.0000 

0.0000 

45000 

0.0035 

0.0033 

40000 

0.0180 

0.0057 

35000 

0.0324 

0.0099 

30000 

0.0551 

0.0203 

25000 

0.0929 

0.0426 

20000 

0.1586 

0.0935 

15000- 

0.2754 

0.2073 

10000 

0.4760 

0.4190 

5000 

0.7730 

0.7151 

surface 

1  .  1485 

1  .  1485 

Since  p  is  greater  than  pa  it  follows  that, 

(782) 


Pi-Po.  Pi  - 

is  smaller  than  — 


pio 


Pao 


and  this  is  compensated  in  the  gravity  equation  by  the  terms 
for  the  circulation  and  radiation.     It  has  been  tested  by  numerous 

p     r> 

computations  that  g  (zi  —  z0)  =  —  Cpa  (Ta  —  T0)  =  —  — 

Pao 

without  important   residuals   from  the  surface  to  the  vanish- 

-p    __   r> 

ing  plane,  and  that  -  Cpu  (Ta  -  T0)  = —    -  holds   true 

pio 

throughout    the    atmosphere.      No     system     of    meteorology 

k 

founded  on  Ra  \  _  ^  =  Cpa  =  constant  is  applicable  in  prac- 
tical studies  of  atmospheric  conditions. 

There  is  another  important  matter  in  which  erroneous  den- 
sities have  been  introduced  into  studies  of  radiation  phenomena. 


ILLUSTRATIONS    OF   THE   USE   OF   ERRONEOUS   DENSITIES    373 

It  has  been  thought  proper  to  modify  the  Bouguer  formula  in 
such  a  way  that  the  observed  variations  in  I,p,  shall  be  dependent 
upon  the  nature  of  the  gaseous  path  m,  so  that  the  integrated 
amount  of  aqueous  vapor  in  the  form  of  liquid  water  may  be 
made  a  factor  attaching  to  the  exponent  m.  Thus  it  has  been 
computed  that  the  water  mass  S/*  =  2eQ,  where  S/*  is  the  integral 
of  the  water  contents  in  grammes  and  CQ  the  vapor  pressure  in 
millimeters  of  mercury  at  the  surface.  The  hair  hygrometer 
measures  the  relative  humidity  of  the  air  at  the  different  levels, 
and  the  corresponding  e  in  mm.  is  computed.  The  full  formula 
is  easily  shown  to  be, 

(783)      M  =  p0f  -f  -  4  (0.622  -I  +  0.235  f). 


It  has  been  customary  to  assume  R  =  287  and  to  omit  the 

e2 

term  in  -57,  so  that  there  remains, 
&*• 

273  6 

(784)      M  =  po-^-X  0.622—. 

This  formula  diverges  from  the  true  one  in  the  same  way  that  p 
differs  from  pa  at  the  different  levels,  so  that  the  sum  of  /*,  S/t, 
is  erroneous  in  the  formula  2/z  =  2e,  being  too  small. 

Furthermore,  the  solar  radiation  suffers  two  sorts  of  deple- 
tion, the  first  as  true  absorption  upon  which  p  depends,  the  sec- 
ond as  scattering  upon  which  /o  partly  depends.  We  have  shown 
how  /o  should  be  modified  for  vapor  contents  (70  —  0.0214  e) 
before  introducing  it  into  the  Bouguer  Formula.  The  ice  crystals 
of  the  upper  strata,  8,000  to  25,000  meters,  are  effective  reflec- 
tors of  the  radiation  energy  in  these  strata.  The  hair  hygrometer 
takes  no  account  of  the  ice  contents  in  the  (3-stage),  nor  the 
water  contents  (f  and  ft  stages)  which  may  be  present,  but 
only  of  the  vapor  contents  (<*-stage).  It  follows  that  since  water 
and  ice  exist  throughout  the  lower  atmosphere  up  to  25,000 
meters,  but  not  always  in  cloud  forms,  though  these  have 
been  observed  up  to  16,000  meters,  that  the  a-stage  Formula 
is  inadequate  to  the  purpose  which  has  been  imposed  upon  it. 


374     EXTENSION  OF  THERMODYNAMIC  COMPUTATIONS 

The  remedy  for  these  difficulties  is  being  considered  in  a  special 
research. 

The  Thermodynamic  Terms 

Table  82  and  Fig.  75  give  the  relations  of  (Qi  -  Q0)  = 
(Wi  -  Wo)  +  (Ui  -  Uo).  (ft  -  Qo)  begins  at  the  surface 
without  value  because  the  external  work  and  the  inner  energy 
are  in  equilibrium;  it  increases  by  a  ci*rve  determining  the 
amount  of  the  lower  absorption  to  a  point  between  the  eleva- 
tions 26,000  and  27,000  meters,  this  being  the  level  where 
2  (Qi  -  Qo)  =  (Ui  -  Uo)  -  (Wi  -  Wo),  which  defines  the  true 
isothermal  level  where  there  is  no  absorption  of  solar  energy, 
in  the  same  sense  that  there  is  no  absorption  at  the  top  and  at 
the  bottom  of  the  atmosphere.  This  gives  three  points  on  the 
line  of  total  solar  energy,  9,806  on  the  vanishing  plane,  about 
5,780  at  26,800  meters,  and  0  at  the  surface.  The  (Ei  -  E0) 
line  of  total  solar  radiation  energy  on  the  several  levels  was 
drawn  by  connecting  these  three  points.  The  area  between  the 
axis  of  ordinates  and  the  (Qi  —  QQ)  curve  represents  the  free 
heat  of  transmission,  the  area  between  the  (Qi  —  Q0)  and  the 
(Ei  —  EQ)  curves  that  of  absorption,  while  (E\  —  E0)  = 
(Qi—  Qo)  +  (Ai  —  AQ),  the  total  area.  The  data  on  the  several 
levels  are  for  1,000-meter  areas,  except  below  3,000  meters, 
where  the  vertical  height  is  500  meters.  The  sum  of  these 
areas  from  the  surface  to  50,000  meters  apparently  represents 
the  true  " solar  constant"  of  radiation  energy,  with  the  constit- 
uents of  transmission  and  absorption. 

The  (Qi  —  Qo)  curve  continues  to  increase  in  value  upward, 
as  (Wi  —  Wo)  and  (Ui  —  U0)  separate,  till  a  maximum  value 
for  the  free  heat  of  transmission  is  reached  at  —  6963  on  the 
38,000-meter  level.  It  will  be  shown  that  the  pyrheliometer 
receives  an  amount  of  heat  at  the  surface  such  as  results  from 
a  summation  up  to  about  this  elevation,  as  if  this  were  the 
general  efficient  source  of  radiation  at  about  1.92  calories. 
The  (Qi  -  Qo)  and  (Ei  -  EQ)  curves  begin  to  diverge  at  27,000 
meters,  and  very  rapidly  after  passing  38,000  meters.  (Qi  -  Qo) 
falls  to  a  minimum  at  48,000°,  and  rises  immediately  to  its 


THE   THERMODYNAMIC   TERMS  375 

primary  maximum,  9,806  at  50,000  meters,  in  this  computation. 
(Wi  —  Wo)  and  (Ui  —  Z70)  pass  through  entirely  similar  changes 
above  38,000  meters,  and  they  all  correspond  to  the  very  rapid 
increase  in  the  temperature  of  the  gases  of  the  upper  absorption 
level.  It  is  evident  that  the  pyrheliometer  measures  the  trans- 
mission and  the  minor  absorption  up  to  about  38,000  meters, 
but  is  not  cognizant  of  the  great  absorption  area  above  it  where 
the  absorption  is  nearly  complete.  The  bolometer  measures  the 
relative  ordinates  of  the  true  solar  spectrum,  unless  some  of 
them  are  absorbed,  but  for  its  total  energy  depends  upon  the 
computations  and  integrations  of  the  spectrum  curve  at  given 
temperatures.  This  is  shown  by  the  spectrum  to  have  lost 
nearly  half  the  incoming  energy  in  the  short  waves  having  long 
ordinates,  these  corresponding  with  the  upper  absorption 
region,  besides  some  minor  absorptions  attributed  to  aqueous 
vapor  selective  absorption  bands  belonging  to  the  lower  region. 
The  general  slope  of  the  (Ei  —  E0)  line  indicates  that  the 
equal  terrestrial*  radiation  is  required  to  balance  the  solar  radi- 
ation, generally  large  below  but  small  above,  in  accordance 
with  the  prevailing  temperatures  and  the  equation  of  equilib- 
rium, as  will  be  explained. 

The  corresponding  values  of  K\Q,  log  c,  a,  in  KIQ  =  c  Ta, 
have  been  computed.  The  exponent  a  =  3.819  at  the  surface 
gradually  falls  to  a  minimum  3.564  at  38,000  meters;  it  rises 
to  a  maximum  3.905  at  45,000  meters,  then  falls  to  about  3,600 
at  the  vanishing  plane.  Further  computations  will  be  under- 
taken in  respect  to  the  region  above  38,000  meters.  (Fig.  79.) 

The  Constituents  of  the  Solar  and  the  Terrestrial  Radiations  in 
the  Earth's  Atmosphere 

It  will  be  convenient  to  transform  the  thermal  data  which 
have  been  computed  in  the  (Meter,  Kilogram-second)  (M.  K.  S.) 
system  into  the  corresponding  values  in  the  (Centimeter-gram- 
second)  (C.  G.  S.)  system,  before  proceeding  to  a  further 
discussion  of  the  transmitted  and  absorbed  constituents  of  the 
solar  and  the  terrestrial  radiations.  The  term  (Qi  —  Q0)  in 


376  EXTENSION  OF  THERMODYNAMIC  COMPUTATIONS 

£o  (Zi  -  80)  =  "  Cpa  (Ta  -  To)  =   -  ?l  ~J*  -  J  fe2  -  g02)  - 

(Ci  ~  Co) 

must  satisfy  the  dimensions  of  all  these  terms,  as  well  as  the 
equations  P  =  PR  T,  and  (ft-  g0)  =  (PFi  -  Wo)  +  (£/!-  tf0). 
(61  -  &)  was  derived  from  -  -  (Cpa  -  -  Cp10)  (Ta  -  T0) 
+  i  (^i2  ~~  ?o2)  =  —  (Qi  —  Qo)  so  that  it  has  the  same  dimensions 

k—  1  P        Pv 

SLS  R  =  —  T  —  Q>,  and  since  R  =  —  ™  =  -~r  they  are  the  same 

as,  those  for  work,  (Wi  —  W0)  =  PW  (vi  —  VQ),  and  for  the 
inner  energy  (Ui  —  U0)  =  Cv  (Ta  —  To).  In  transformations 
from  the  (M.  K.  S.)  to  the  (C.  G.  S.)system  the  dimension  factors 
are,  [M]  =  1000  =  103,  [L]  =  100  =  102;  in  transformations 
from  the  (C.  G.  S.)  to  the  (M.  K.  S.)  system,  [M]  =  1Q-3, 
[L]  =  10~2.  All  are  in  mechanical  units. 


Pressure,  kilog./met.»,  [P1-Po]=ML~i 


_  _ 

Density,  kilog./met.»,  [ft0]        =ML~M        P10        ~  'L 
Velocity  square,  [gi2  —  g02]         =  i2  =   104   =  ....................  =  10000 

Specific  heat,  [Cp.Cv.R.K.]     =  U  =  104  =  ....................  =  10000 

Heat,  [<2i  -  <2o  =  (C£«  -  Cp10)  (Ta  -  Tc)]   =  L2  =  104  =  ........  =  10000 

Work,  [(TFj  -  TFo)  =  P10  (»i  -  »o5]  =  ML~\  M~!  L3  =  U  =  104  =  .  =  10000 
Inner  Energy,  [(Z7t  -  Z70)  =  Cy  (r0  -  T0)]  =  L2  =  104  =  ........  =  10000 

The  common  factor  for  reducing  all  the  terms  of  the  gravity 
and  heat  equations  from  (M.  K.  S.)  to  (C.  G.  S.)  in  mechanical 
units  is  104  =  10,000. 

From  the  general  equations  (Wi  —  Wo)  =  PW  (VI—VQ)  and 
(Qi  ~  Qo)  =  (W1  -  W0)  +  (£/!  -  J70),  we  have, 

U,  -  U0      ft-Qo  a 

AIO  =  "•          -   =  —  -  —  X       =  C  L 


—  V0 


which  is  the  radiation  equation  employed  in  the  computations. 
The  dimensions  of  all  these  terms  are  the  same  in  reducing  from 

10000 
the  (M.  K.  S.)  to  the  (C.  G.  S.),  the  factor  being  -       -  =  10,  as 

1UUU 


SOLAR  AND   TERRESTRIAL  RADIATION  377 

is  commonly  used  in  the  pressure  P1Q.  The  reduction  from 
mechanical  units  to  heat  units  (C.  G.  S.)  is  by  the  factor 

4  1851  X  107'  and  the  factor  for  chanSe  from  second  to  minute 
is  60.  Hence,  to  reduce  (Qi  —  QQ)  in  M.  K.  S.,  mechanical  units 

gram  calorie          , 

into  -— — — ,  the  factor  is, 

cm.2  minute 

1000^1851%  -  Sil5  =0-^14336;  Log.-5.15644. 

Compare  the  constants,  coefficients,  and  dimensions  of  Table  95, 
where  the  formulas  afford  many  other  combinations  in  conform- 
ity with  the  kinetic  theory  of  gases. 

The  Total,  Transmitted  and  Absorbed  Amounts  of  the  Solar  and 
the  Terrestrial  Radiations 

Huron,  September  i,  1910 

The  method  of  separating  the  total  radiation  (Ei  —  E0) 
received  at  the  earth  from  the  sun  into  the  transmitted  (Qi  —  Q0) 
and  the  absorbed  (Ai  —  A0)  constituents  on  each  level  has  been 

explained.      The  coefficient   of    transmission  is  p  =  ~ ^=°, 

£L\  ~  £L>Q 

and  of  absorption  is  k  =  ~ ^r.    The  terrestrial  radiation 

h,\  —  £Lo 

is  obtained  as  follows :  Let  JQ  —  c  T4,  the  black  body  radiation 
of  the  atmosphere  at  the  point  whose  temperature  is  T;  let 
Ja  =  c  Ta  the  actual  radiation  for  c  and  a  as  computed  from 
the  thermodynamic  data,  so  that  the  actual  absorption  between 
two  planes  is  /a-i  —  Ja-o  =  c\  T\l  —  CQ  TQa°.  The  general 
equation  of  radiation  equilibrium  is 

2  J0  =  E  +  D,  where  D  is  the  total  terrestrial  radiation. 
Hence,  D  =  2  JQ  —  E,  and  B  =  D  —  Ja,  where  B  is  the  trans- 
mitted terrestrial  radiation.  The  coefficient  of  terrestrial 

7?    -  -  7? 
transmission  is  p1  =  77 ^,  and  that  of  absorption  is  k1  = 

U\  —  L/o 

~^ pp-0.    The  results  of  these  computations  appear  on  Table 

— 


378 


EXTENSION   OF   THERMODYNAMIC   COMPUTATIONS 


w    o 

rt     rH 


u     2 

§  « 

£    c/) 


s    ^ 

IcS 


<X    § 

8  I 

•<3  O 
H  Jo 
g  ^ 

I  I 
"  I 

B  § 

I5 

a 


*    2 


'31 


1« 

'J3   S 


CO 
CN 


CO  CO  CO  CO  CO  CO  CO 
CO(MOOOSCOCOCsQ 
OOS'*ttt>-  OSO5OSO01>-COiOCO 
CO  CO  i—  t         rHCO»OiOCOcOCOcOcOCOCOcOcOCDcOcO»-OiO 

I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I 


^J^  CO 

O:ll>» 
rHOSOO 


rH    CO    rH    OS    O    CO 
I         I         I         I         I 


CO  O  Tt<  CO  O  CN 
*H  O  00  £-  CO  00 
OS  Tt<  l>  l>  l>  l> 


00  10  00  1> 


I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I 


COCOOOt^OOOSOOCOOOO 

lO(MOSTtlrHI>.COrHlCOb- 
rHrHrHrHrHOOOOpp 

r  r 


rHCOb-OSOTh'OOC^OOrHlOCOOOCOOOrHlOGOCMCO 

I   I   I   I   I   I   I   I   I   I   I   I   I   I   I   I   I   I   I   I 


I>-OTH|>.O'~"i 

OSl— lO5O»Ol— I 

»O^lCNC^ir-i'— I 


rH    CO 
CN    rH 


CO  *O  CO  *O  CO 


SCOOO5Tt<rHTt<COrHOOSb-TtlGO 
OS*OGOrHCOTt<lOCOb-b-OOOSOS 

!>•  b*  OO  00  Os  Os  Os  Os  Os  Os  OS  Os  OS  os 


rH   OCJ    O   b-   »O   CO 
rH    rH    rH    O   O   O 


O  O 


00  00 


I   I   I   I   I   I   I   I   I   I   I   I   I   I   I   I   I   I   I   I   I   I  + 


CO   <M    OS   OS   CO 


O  CO  CO  00  00 

IO    CO    rH  OS    b- 

OS  OS  OS  00  00 

o  o  o  o  o 


OOOlOCNOI^^fNOSb-TtHrHOSCOT^rHGOCOCOrHOOJOCO 
«tfCOCOCOCO<N<N<NrHrHrHrHOOOOO5O5OSOSOOOOOO 
^^1— IrHrHrHrHrHrHrHrHrHrHrHrHrHOOOOOOO 

r  r  r  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \  \ 


SOLAR   AND    TERRESTRIAL   RADIATION 


379 


CMrHCSt^-tf't^^t^OOiOt^cOr-  icDCOfNiO^COCOTttcOC^^iOOO-^ 
COTfi—  i  OS  l>  -^  r-  (OOiO<NOOiO<NOOcOTficN'-HOO5001>t>'COCN>-i 
uooO^T^T*Tt<COCOCO<NcN<NrHrHrHrHrHrH 

I   I   I   I   I   I   I   I   I   I   I   I   I   I   I   I   I   I   I   I   I   I   I   I   I   I   I   I 


I   I 


' 


b-  OO 

CO  rH 
^H  <M 


OOOOCO   •  rfi 
rH  OS  OS   •  i—  i 


•  •<NOOO5t»»-ia500l> 

•  •  <N  CO  OS  i—  i  >—  i  CO  GO  00 


i-H  CO       •       • 

l>*      "^ 

i-H    <N         •         . 


CO  (M  O  i-4   •  i-H 
CO  GO  O  OS   -00 


- 

^^  rH  T-H  ^^  i-H  T-H  T-H  C^J  rH  rH  C^  rH  CO  CO  CO  "^  CO  ^O  t^»  00  t^*  ^^  OS  CO  ^  f^  IO  ^O 

CO  CO  CO  CO  CO  CO  CO  CO  CO  CO  CO  CO  CO  CO  CO  CO  CO  CO  CO  CO  CO  i"H  ^^  TH  CO  CO  CO  CO 

I   I   I   I   I   I   I   I   I   I   I   I   I   I   I   I   I   I   I   I   I   I   I  I  I  I  I   I 


'  iO 

:  b- 


§  :s 


+++ i ++++ i i i i i i i i i i i i i + i i i i 


T— (Ol>-TtiOO5rHrHOCO 
OOOOi— i  i— iCOCO^^f 


.  CO 

•  I 


.  00 

;  oo 


01 


rH    CO    CO    O 
t^    rH    l^    rH 

00  00  l>  l~~ 


8888 


+  +     I  I  i  1  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I 


CO  CO  CO 


rt<  b»  CO  00  l^  00 

rH    CO    rH    CO    rH    CO 


GO    CO    Tt<    CO    CM    rH    O 

o  o  o  o  o  o  o 


8 


cooooco 

^f  "^  ^t^  CO  CO 


r  r  i  i  r  i  i  i  i  i  i  i  i  i  i  i  i  i  i  i  i  i  i  i  i  i  i  i  i 


t^-f>-ix*t>»r>-iv 

d  C^l  Ol  i-^  ^H  i-^ 


CM  <N  <N  <N  <N  <N  CN 


»o  o 


380  EXTENSION   OF   THERMODYNAMIC   COMPUTATIONS 

84,  inj^'a^  units.     The  values  of  (Ql  -  Q0)  in  M.  K.  S. 

mechanical  units  are  found  in  the  last  column.  The  summa- 
tions are  as  follows  for  the  entire  depth  of  the  atmosphere. 

Total  sum  of  solar  radiation,    2  (Ei    -  £0)    =  -  3.6942  gr\ Ca.' 

cm.2mtn. 

Transmitted  solar  radiation,  2  (Qi  -  Q0)  =  -  2.7177  " 
Absorbed  solar  radiation,  S  (Ai  —A0)  =  -0.9101  " 
Atmospheric  (black)  radiation,  S  (/O.I-/O.Q)  =  -  3.8181  " 
Total  terrestrial  radiation,  S  (A  -A)  =  -  3.9419  " 
Transmitted  terrestrial  radi- 
ation, S  (Bi  -B0)  =  -  2.5744  " 
Absorbed  terrestrial  radiation  2  (Ja>i— Ja-o)  =  —1.3675 

The  minus  (— )  sign  is  due  to  the  fact  that  while  the  positive  (+) 
direction  is  along  the  axis  z  outward,  the  temperature  and 
thermal  gradients  decrease  in  this  direction;  a  positive  (+) 
sign  would  indicate  an  inversion  of  temperatures  and  thermal 
quantities  outward.  In  the  case  of  p,  k,  the  coefficients  are  all 
positive;  in  the  case  of  pl,  k1  they  are  generally  positive  in  the 
lower  levels,  but  sometimes  (+)  and  sometimes  (  — )  occur 
in  the  upper  levels.  Since  the  original  observations  of  the 
velocity  are  lacking,  those  used  being  supplied  by  analogy,  it  is 
probable  that  the  data  of  Huron,  September  1,  1910,  are  inade- 
quate to  produce  accurate  values  of  pl,  k1. 

II.  SECOND  DISTRIBUTION  OF  TEMPERATURE 
The  Balloon  Ascension,  Uccle,  November  9,  ign 
As  already  stated  the  gravity  residuals  in  the  check  equation 

(196)        g  (z,  -  zo)  =  -  Pl~P°  -  (cPa  -  C#io)  (Ta  -  T0) 

Pio 

+  A  g  (Zl  -  z0) 

were  quite  large  above  40,000  meters  in  the  Huron  ascension. 
It  was  supposed  that  this  was  due  to  the  fact  that  the  velocities 
q  were  assumed,  that  g  was  taken  constant  in  the  adiabatic 


SECOND    DISTRIBUTION    OF    TEMPERATURE  381 

gradient,  and  that  the  formulas  might  begin  to  fail  when  the 
P,  p  terms  became  very  small.  Accordingly,  a  new  series  of 
computations  has  been  undertaken  for  Uccle,  June  9,  Septem- 
ber 13,  November  9,  1911,  the  necessary  data  including  veloc- 
ities having  been  courteously  supplied  by  Dr.  Vincent,  Directeur 
de  1'Institut  Royal  Meteorologique  de  Belgique.  In  these, 
besides  using  the  observed  velocities,  the  gravity  ranges  with 
the  height.  It  is  known  that  a  temperature  T  introduced  into 
the  series  of  computations,  building  upward  from  the  surface 
from  level  to  level,  whether  observed  or  assumed,  can  always 
be  checked  by  the  control  gravity  equation.  If  the  series  of  T 
is  correct,  A  g  (^  —  z0)  vanishes.  Hence,  it  is  practical,  by  a 
set  of  trial  computations,  to  arrive  at  such  temperatures  through- 
out the  atmosphere  as  will  satisfy  the  entire  group  of  ther- 
modynamic  equations-.  The  balloon  ascensions  to  20,000  or 
30,000  meters  give  the  necessary  foundations  upon  which  the 
entire  structure  can  be  built  up  to  90,000  meters.  Up  to  40,000 
meters  the  four  ascensions  give  about  the  same  residuals,  with 
occasional  wide  pairs  depending  on  an  erroneous  T.  The  trials 
to  reduce  the  check  residuals  were  therefore  limited  to  the  region 
above  40,000  meters.  It  appears  that  positive  residuals  are  to 
be  diminished  by  increasing  the  temperatures  on  the  several 
levels.  The  general  outcome  was  to  introduce  a  second  tem- 
perature region  beginning  at  about  T  =  170°  on  the  40,000- 
meter  plane  and  terminating  near  the  90,000-meter  plane.  The 
remarkable  result  is  that  the  formulas  are  rigorous,  however 
small  the  values  of  P,  p,  R,  may  become  for  the  successive  T, 
and  that  small  residuals  above  50,000  meters  imply  rather  large 
changes  in  the  temperatures  T.  Extending  the  temperatures 
from  40,000  meters  to  levels  higher  than  50,000  meters,  the 
residuals  have  the  following  mean  values: 

Huron,  Sept.     1,  1910,  40,000  to  55,000,  three  trials,  A  g  (zi-z0)  =  +163.0 

Uccle,    June     9,  1911, 40,000  to  69,000,  four        "  "  =  +  17.8 

Uccle,     Sept.  13,  1911, 40,000  to  80,000,  four        "  "  =  +     4.8 

Uccle,     Nov.    9,  1911,  40,000  to  90,000,  two  =  +     2.5 

Table  85  contains  a  summary  of  the  Uccle,  Nov.  9,  1911, 
ascension,  where  the  gravity  differences  may  be  examined  in  the 


382  EXTENSION   OF   THERMODYNAMIC   COMPUTATIONS 


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THE   THERMODYNAMIC   TERMS 


383 


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384  EXTENSION   OF   THERMODYNAMIC   COMPUTATIONS 

sixth  column.  They  are  smaller  from  90,000  to  65,000  than 
from  65,000  to  40,000  meters.  The  inference  is  that  the  entire 
thermodynamic  system  up  to  vanishing  quantities  is  reliable  for 
numerous  researches  depending  upon  the  data.  In  order  to 
illustrate  how  small  these  quantities  really  are,  we  have  at 
65,000  meters,  B  =  0.001225  millimeter  of  mercury,  p  = 
0.00000009629  C.G.S.,  while  hydrogen  at  normal  surface  con- 
ditions is  pH  =  0.00008924  grJcm3.  The  remarkable  precision 
of  the  computations  is  proved  by  the  system  of  checks  from  the 
surface  to  90,000  meters. 

These  data  above  40,000  meters,  therefore,  modify  the  data 
of  Table  82,  and  may  be  substituted  for  them.  The  results 
of  further  studies  on  these  computations  will  be  published  in 
Bulletin  No.  4  of  the  Argentine  Meteorological  Office.  It  may 
be  here  remarked  that  the  thermal  efficiency  of  the  atmosphere 
begins  at  about  65,000,  there  being  little  absorption  above  it. 
The  shape  of  the  absorbing  area  above  40,000  meters  has  been 
changed  by  this  extension,  but  the  amount  is  not  very  different, 
as  will  be  illustrated.  Compare  Figs.  74,  75,  79. 

Summary  of  the  Computations  for  Twenty-one  Balloon  Ascensions 

In  order  to  improve  the  data  as  far  as  practical,  similar 
computations  were  extended  to  twenty  other  balloon  ascensions. 

(10)  United  States,  Omaha,  February  21,22,23,  1911;  Sep- 
tember 28,  1909.  Huron,  September  1,  4,  7,  16,  1910.  In- 
dianapolis, October  6,  30,  1909. 

(6)  Europe,  Lindenburg,  July  27,  1908;  April  27,  May  5,  6, 
September  2,  1909.  Milan,  September  7,  1907. 

(5)  Atlantic  Tropics,  Otaria,  June  19,  1906;  July  29,  August 
29,  September  9,  25,  1907.  The  mean  values  for  the  several 
quantities  appear  on  Table  86,  where  the  data  can  be  conveniently 
examined. 

(Ei  —  Eo).  The  same  values  were  adopted  throughout, 
but  since  balloon  ascensions  now  reach  28,000  meters  the  critical 
value  at  that  elevation  can  be  further  examined. 

(61  -  60),  (Ai  -  40).    These  areas  are  seen  on  Fig.  75,  and 


SUMMARY  FOR  TWENTY-ONE  BALLOON  ASCENSIONS         385 

they  have  been  described.  It  should  be  noted  that  the  course 
of  (Qi  -  Qo)  in  the  lower  absorption  region  does  not  in  the 
least  follow  the  distribution  of  the  aqueous  vapor  contents, 
which  is  at  a  minimum  where  (Qi  —  Q0)  is  at  its  maximum.  This 
problem  is  very  difficult  to  solve  satisfactorily. 

p  increases  rapidly  up  to  2,000  meters,  then  more  slowly 
up  to  p  =  1.000  in  the  true  isothermal  level  at  27,000  meters; 
above  that  level  p  falls  to  a  small  value  p  =  0.053  at  46,000 
meters,  and  then  rises  to  about  unity  on  the  vanishing  plane. 

k  passes  through  inverse  relations  in  respect  of  p. 

JQ  has  a  maximum  value  at  4,000  meters,  and  proceeds 
irregularly  to  zero  at  the  top.  Its  value  can  always  be  recovered 
from  J0  =  $  (D  +  E)  in  the  tables. 

D  has  a  large  negative  maximum  at  the  surface,  passes 
through  zero  with  change  of  sign  near  18,000  meters,  and  gradu- 
ally increases  to  its  maximum  +  0.1406  on  the  vanishing  plane. 

B  falls  from  a  negative  maximum  at  the  surface  to  zero  at 
the  13,000-meter  level,  at  the  bottom  of  the  so-called  isothermal 
layer,  and  then  increases  irregularly  to  the  top  of  the  atmosphere. 

Ja  begins  with  a  maximum  at  the  surface  and  gradually 
falls  to  a  small  quantity,  finally  vanishing  at  the  top. 

p1  has  a  nearly  constant  value  of  0.800  in  the  convectional 
region  and  probably  about  0.500  above  it.  kl  is  correlative  to  p1. 

Uccle,  November  9,  1911 

The  data  are  fully  computed  for  the  Uccle  ascensions,  as 
extended  to  90,000  meters,  and  the  results  above  40,000  meters 
will  be  briefly  summarized.  The  total  atmospheric  radiation 
energy, 

I  Jo  =  2  (Cl  TV  -  c0  TV) 

from  the  surface  to  90,000  meters,  is  equivalent  to  the  "solar 
constant"  at  the  distance  of  the  earth,  because  it  represents  the 
amount  of  heat  required  to  maintain  the  existing  temperature 
distribution  in  equilibrium  with  the  incoming  and  outgoing 
radiations  that  are  in  operation  day  and  night.  Summarizing 
the  Jo  data,  we  have, 


386  EXTENSION    OF    THERMODYNAMIC    COMPUTATIONS 


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SUMMARY  FOR  TWENTY-ONE   BALLOON  ASCENSIONS         387 


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388 


EXTENSION  OF  THERMODYNAMIC  COMPUTATIONS 


'  66,000 

'  76,000 

"  56,000 

'  66,000 

"  46,000 

1  56,000 

"  36,000 

'  46,000 

"  26,000 

'  36,000 

"  16,000 

'  26,000   ' 

"   6,000 

'  16,000   ' 

"  Surface 

'  6,000 

From  the  76,000  to  90,000  meters,  2  J0  =    -0.0001 

-0.0032 
-0.0372 
-0.1981 
-0.2952 
-0.2914 
-0.3275 
-1.0151 
- 1 . 5658 

Total  atmospheric  radiation  energy       —3.7336 

From  the  Huron  ascension  we  obtained  —3.8181.  It  is 
apparently  a  question  of  distribution  rather  than  the  amount 
that  is  concerned  in  these  two  temperature  systems.  The  total 
" solar  constant"  derived  from  the  black  atmospheric  radiation 
is  about  twice  the  amount  measured  by  the  pyrheliometer.  More 
data  on  this  important  subject  will  be  accumulated. 

First  Method  of  Computing  the  Pyrheliometric  Data. 
The  Effective  Energy  of  Radiation  and  the  Solar  Constant. 

When  the  preceding  Chapter  V,  on  radiation,  was  written  it 
was  supposed  that  the  extension  of  these  computations  would 
be  limited  to  the  lower  levels  in  the  atmosphere  where  direct 
temperatures  were  obtained  in  balloon  ascensions.  But  'it  is 
now  evident  that  temperatures  may  be  measured,  or  assumed 
by  trial,  and  they  will  be  correct  provided  they  produce  such 
values  of  P.  p.  R.  T.  in  successive  stages  as  will  satisfy  the 
gravity  equation  with  small  residuals, 

Pi  -Po 

(196)      g  (zi  -  ZQ)  =  -  -  J  (qi2  -  £02)  -  (Qi  ~  Co). 

Pio 

Hence,  by  a  series  of  trial  computations,  it  has  become 
possible  to  extend  the  data  up  to  90,000  meters,  and  four  balloon 
ascensions  have  been  computed  up  to  70,000  or  90,000  meters. 
We  thus  obtain  by  (335),  (333) 
(Qi  ~  Co)  =  (Wi  -  W0)  +  (Ui  -  U0)  =  P10  (vi  -%)  +  (tfi-  Z/o), 


=  K 


-Co 


=  Ja 


for  selective  radiation,  and 


THE   CONSTANTS  AND   COEFFICIENTS  FOR  DRY  AIR 


389 


(344) 


J0  =  ci  2V  -  c0  TV 


for  black-body  radiation,  in  each  1,000-meter  stratum  from  the 
surface  upward,  and  thence  the  total  thermodynamic  and 
radiation  energies  throughout  the  atmosphere  by  taking  the 
summations  for  all  of  the  strata.  The  results  are  summarized 
in  Tables  82,  85,  and  Figs.  75,  79.  Similarly,  the  computations 
include  the  data  of  the  Kinetic  Theory  of  Gases,  (H9U)  external 
and  internal  energies;  (q.  f)  velocities;  (n.  N)  number  of  mole- 
cules, lmax  free  path  length,  y  number  of  collisions,  WH  mass  of 
the  hydrogen  atom,  e^  the  negative  ion  charge,  in  all  strata  to 
90,000  meters.  Tables  96,  97.  See  BuUetin  No.  4,  O.  M.  A. 

TABLE  87 

SUMMARY  OF  THE  RESULTS  OP  THE  COMPUTATIONS  ON  THREE  BALLOON 
ASCENSIONS  FOR  THE  VALUES  OF  THE  THERMODYNAMIC  AND  RADIATION 
ENERGIES 


Station  and  Date  of  the  Observations 
Height  in   Meters  of  0°   Temperature 

Uccle 
June  9,  1911 
70,000 

Uccle 
Sept.  13,  1911 
80,000 

Uccle 
Nov.  9,  1911 
90,000 

2%(zi—  20)]  gravity  acceleration  .... 
2[—  (£<?i2—  <7o2)]     kinetic    energy    of 
circulation    ... 

9.5792 
0.0000 
5.4250 

4.1542 
4.1146 

3.9690 

11.0895 
-0.0006 
6.6676 

4.4219 
4.1608 

3.9261 

12.4565 
0.0002 
8.6590 

3.7975 
3.9386 

3.5373 

2[  —  (Qi  —  Qo)]    free  heat   (non-adia- 
batic)              

S[0(2i-2o)  +M?i2-?o2)  -f- 
(Qi-Qo)]  summary  

L               PlQ       J      " 

S[/o  =  CiTV  -  CoTV]  black  body  ra- 
diation   

2  [K  i  —  KO]  radiation  energy             .  . 

1.4528 

0.0000 
1.4602 

1.4603 

1.4672 

-0.0087 
1.4404 

1.4600 

1.4802 

0.0352 
1.4085 

1.4395 

t 

2  rQi  —  Qo~|  free  heat  per  volume 
l_  v\.  —  #o  J  change 

2  [Pi—  Po]  pressure  differences  
2[/0  =  CiTial  -  c07>0]  selective  ra- 
diation   

General  mean  of  the  thermodynamic  data 
General  mean  of  th&black  body  radiation  data  /0 
General  mean  of  the  radiation  and  pressure  data  . 
General  mean  of  the  selective  radiation  data  Ja    . 


4.0979 
3.9476 
1.4516 
1.4536 


390 


EXTENSION    OF    THERMODYNAMIC    COMPUTATIONS 


Table  87  shows  that  the  thermodynamic  state  of  the  earth's 
atmosphere  is  such  that  about  4.00  gr.  cal./cm.2  min.  is  required 
to  raise  it  from  the  frozen  and  solid  state  of  air  at  0°  A  into 
the  expanded  state  it  now  has  with  its  P.  p.  R.  T.  in  all  strata. 
This  is,  also,  shown  to  be  equal  to  the  black  body  radiation  JQ 
as  derived  from  the  thermodynamic  data.  The  selective  absorp- 
tion Ja  in  a  state  for  continuous  emission  is  about  1.46  calories, 
and  this  is  equal  to  the  temporary  amount  of  energy  in  the 
(K!  -  KQ)  and  (Pi  -  Po)  terms. 

Table  88  contains  a  summary  of  the  terms  in  the  equation  of 
equilibrium,  (785)  S  =  /i  +  B  +  Ja  and  h  =  R. 


TABLE  88 

THE  TERMS  IN  THE  "EFFECTIVE"  RADIATION  FORMULA, 

WHERE  h  =  R 


—  R  —  Ja 


z 

5 

Ja 

h  =R 

P 

Pw 

Wo 

90000 

4.00 

0.000 

2.000 

1.000 

14000 

76000 

4.00 

0.000 

2.000 

1.000 

10000 

66000 

4.00 

0.000 

2.000 

1.000 

10000 

56000 

4.00 

0.000 

1.999 

0.999 

10000 

46000 

4.00 

0.007 

.993 

0.998 

10000 

36000 

4.00 

0.034 

.983 

0.996 

6000 

36000 

4.00 

0.070 

.967 

0.994 

.  .  . 

3000 

27000 

4.00 

0.100 

.950 

0.992 

. 

3000 

24000 

4.00 

0.130 

.935 

0.991 

3000 

21000 

4.00 

0.170 

.915 

0.988 

3000 

18000 

4.00 

0.220 

.890 

0.984 

3000 

15000 

4.00 

0.295 

.853 

0.975 

3000 

12000 

4.00 

0.410 

.795 

0.961 

3000 

9000 

4.00 

0.580 

.710 

0.950 

3000 

6000 

4.00 

0.776 

.617 

0.929 

0.920 

3000 

3000 

4.00 

1.050 

.475 

0.887 

0.890 

3000 

000 

4.00 

1.461 

.270 

0.840 

0.840 

/i  =  the  "effective"  radiation  =  2.00  at  90,000  meters. 

S  =  the  solar  constant  =  4.00  calories. 

R  =  the  " reflected"  radiation  =  2.00  at  90,000  meters,  and 

neutralizes  one-half  the  solar  constant,    /i  =  R  on  all 

levels,  except  during  changes  in  T. 
Ja  =  the    absorbed  radiation  in  the  lower  levels,   small  at 

40,000  meters,  and  increasing  by  variable  values  of  p 


THE  EFFECTIVE  ENERGY  OF  RADIATION        391 

in  the  Bouguer  Formula,  for  mo  about  3,000  meters, 

/  =  /.#-.• 

p  =  the  values  of  the  coefficient  of  transmission  formed  by 

p  =  —  using  the  values  of  7  for  different  3,000-meters. 

lo 

pw  —  the  general  observed  values  of  p  at  the  sea  level,  at  3,000 
meters   as  La  Quiaca,  with  its  probable  value  at  6,000 
meters. 
m0  =  the  adopted  depth  of  the  stratum  in  the  zenith. 

It  is  evident  that  p  is  not  constant  throughout  the  atmos- 
phere, and  that  mo  is  not  to  be  taken  as  unity  for  the  90,000 
meters.  On  the  other  hand, 

7T      ^  m>         ^  m2         A.  m*  x    1*n 

=  /O  pl       •   p2       •  p3  •       •       •      pn      , 

for  variable  (p.  m.). 

/i=  the  effective  radiation  as  observed  by  the  pyrheliometer. 

/o  =  2.00  cal.  at  90,000  meters, 
/o    penetrates      to     30,000     meters    undepleted,    and    then 

diminishes  by  a  variable  p,  constant  through  m  =  3,000 

meters,  in  the  Bouguer  Formula,  to  1.46  calories  at  the 

surface. 
R  =  /i  at  every  level.     This  return  current  neutralizes  one-half 

of  the  solar  constant  S  =  4.00  calories. 
Ja  =  the  absorbed  energy  in  the  lower  levels,  and  together 

with  R  neutralizes  J  S. 

The  Bolometer  measures  the  transmitted  parts  of  S. 
The  Pyrheliometer  measures  the  transmitted  parts  of  70, 
that  is,  one-half  the  "solar  constant." 

Figure  76  shows  the  relations  of  these  terms  throughout  the 
atmosphere.  The  effective  radiation  /o  proceeds  downward 
till  it  is  deflected  by  Ja  acting  outward,  arriving  at  the  surface 
as  1 1.  Simultaneously,  the  returning  radiation  R  =  I\  in  every 
level,  during  temperature  equilibrium,  reaches  the  top  of  the 
atmosphere,  with  the  addition  of  Ja  in  the  lower  levels, 
having  the  value  R  =  2.00  cal.  which  neutralizes  one-half 
of  the  incoming  S  =  4.00.  It  is  easily  seen  that  the  pyrheli- 


392  EXTENSION  OF    THERMODYNAMIC   COMPUTATIONS 

E  =  1.950  S  =  3.950 


0 
I  0.000 


0.500          1.000  1.500 

Scale  for  the  density  p    1.200 


2.000 
0.960 


2.500 
0.720 


3.000 
0.480 


3.500 
0.210 


4.000 
0.000 


FIG.  76.    Two  methods  of  discussing  the  pyrheliometer  observations. 


p 

A 

B 

Area 

Area 

Area 

Ja 

J0 


=  the  solar  constant  3.95  gr.  cal./cm.2  min. 

=  the  radiation  energy  absorbed  in  producing  temperature  T. 

=  the  density  curve. 

=  the  probable  pyrheliometer  curve. 

=  the  probable  bolometer  curve. 

S  —  B   =  the  temperature  radiation. 

B  —  A  =  the  scattered  radiation. 

A  —  0  =  the  free-heat  radiation. 

absorbed  energy  in  1000  meters. 

black  body  energy  in  1000  meters. 

intensity  of  zenith  sun. 


=  I  \  /  pa- 


THE     EFFECTIVE    FORCE     OF     RADIATION 


393 


ometer  measures  only  the  effective  radiation  70  =  J  S,  while 
the  bolometer  measures  such  ordinates  as  are  necessary  for  the 
construction  of  a  4.00  calories  curve  at  solar  temperature  be- 
tween 6,900°  and  7,000°. 

If  we  supply  by  simple  interpolation  the  missing  ordinates 
on  the  energy  spectrum  curve  for  6,900°,  so  that  the  wave-length 
interval  is  A  A  =  0.05  /*,  and  similarly  fill  out  Abbot's  ordinates 
for  Washington,  D.  C.  (34  m),  Mt.  Wilson  (1,780  m),  Mt. 
Whitney  (4,420  m),  (Bui.  No.  3,  O.  M.  A.  Tables  XXVI, 
XXVII),  we  have  the  following  results  for  the  sum  of  the  ob- 
served ordinates,  that  is,  the  relative  areas  of  transmission. 

TABLE  89 

COMPARISON  OF  THE  ABSORBED  ENERGY  BY  THE  BOLOMETER  AND  THERMO- 

DYNAMIC  DATA 


Energy  Spectrum 

2J 

D 

t 

Ratio 

Ja 

fci 
Ratio 

6900°  

80.295 

4.00 

Washington  

51.19 

29.105 

0.363 

1.45 

0.363 

(34) 
Mt.  Wilson  

59.55 

20.745 

0.258 

1.15 

0.287 

(1780) 
Mt.  Whitney  
(4420) 
Final  curve   .  .    . 

61.38 
67.26 

18.915 
13.035 

0.236 
0.162 

0.90 
0.65 

0.225 
0.163 

(8000) 

S  /  is  the  sum  of  the  ordinates  of  transmission  at  each  station. 

D  is  the  absorbed  part  =27  (6,900°)  -  S  /  (station). 

k  is  the  ratio  of  absorbed  part  at  each  station  to  £  J  (6,900°). 

Ja  is  the  computed  thermodynamic  absorption  at  same  levels. 

k1  is  the  ratio  of  the  absorbed  part  at  each  station  to  4.00. 

It  is  seen  that  these  two  ratios  (k,  k1)  are  substantially  the 
same,  except  for  minor  variations. 

It  must  be  concluded  that  the  amount  of  absorption  in  the 
lower  atmosphere  Ja  is  about  the  same  as  the  amount  of  deple- 
tion in  the  bolometric  energy  spectrum,  referred  to  a  6,900° 
curve,  or  4.00  calories  for  the  solar  constant.  This  analysis 
points  to  a  very  different  method  of  discussing  pyrheliometer 


394     THE    EXTENSION    OF    THERMODYNAMIC    COMPUTATIONS 

observations  from  that  commonly  practised,  and  it  promises  to 
harmonize  the  several  branches  of  this  hitherto  conflicting 
subject. 

SUMMARY 

1.  The  bolometer  curves  as  observed  are  best  satisfied  by 
an  energy  spectrum  of  6,900°  or  a  solar  constant  of  4.00  gr.  cal./ 
cm.2  min. 

2.  The  thermodynamics  of  the  atmosphere  requires  an  ex- 
penditure of  4.00  calories  to  produce  and  maintain  the  existing 
pressures,  densities,  temperatures,  and  thermodynamic  values 
up  to  90,000  m. 

3.  The    pyrheliometer    measures    only    one-half    the    solar 
constant,  that  is,  the  efficient  energy  2.00  calories,  because  one- 
half  of  the  incoming  radiation  is  neutralized  by  the  returning 
energy  stream  of  2.00  calories. 

4.  The  amounts  of  absorbed  radiations,  as  measured  by  the 
bolometer  ordinates  or  by  the  thermodynamic  conditions,  agree 
in  giving  about  the  same  ratios  (k,  k1). 

5.  The  conclusions  derived  from  the  extensive  computations 
summarized  in    this    section    verify,    generally,    the    text    in 
Chapter  V. 

Change  of  Theory 

The  foregoing  analysis  is  based  upon  the  view  that  the 
pyrheliometer  measures  the  efficient  incoming  solar  radiation  at 
2.00  calories,  and  upon  the  formula  that  half  the  incoming  ray 
advances  to  the  surface  while  half  of  it  is  scattered  back  to  space. 
Further  experience  brings  both  these  ideas  into  doubt,  and  we 
proceed  to  give  evidence  that  the  I0  —  curve  of  Fig.  76  should 
be  made  the  Ia  —  curve,  thus  enclosing  the  scattering  of  radia- 
tion between  the  curves  A  and  B. 

The  Second  Method  of  Discussing  Pyrheliometric  Data 

The  discussion  of  pyrheliometer  observations  begun  in 
Chapter  V  has  been  continued  by  the  development  of  a  new 


A  NEW  METHOD  OF  DISCUSSING  PYRHELIOMETRIC  DATA         395 

method  of  computation,  which  will  be  briefly  described.  Com- 
pute from  the  observed  pyrheliometer  readings  the  values 
of  the  intensity  /i,  for  the  sun  in  the  zenith,  and  with  pw) 
the  unconnected  coefficient  of  transmission,  extrapolate  I0  the 
intensity  reduced  to  the  mean  solar  distance.  Abbot  multiplies 
I0  by  a  bolometer  factor,  1.123  for  Washington,  and  1.094  for 
Mt.  Wilson  on  the  average,  to  which  a  small  correction  is  then 
added  for  the  effect  of  the  aqueous  vapor  pressure  e.  We  pro- 
ceed to  develop  the  data  in  another  way.  Collect  the  individual 
observations  in  convenient  groups,  according  to  the  observed 
values  of  pw,  as  (0.900  -  0.880),  (0.880  -  0.860),  ...  for 
IOJ  /i,  6j  and  take  the  mean  values,  such  as  appear  in  Table  90 
for  Washington,  D.  C.,  at  34  meters,  Bassour  at  1,160  meters, 
La  Quiaca  at  3,465  meters.  Plot  on  diagrams  with  pw  for 
abscissas,  and  values  of  70,  /i,  in  gr.  cal./cm.2  min.  for  ordinates. 
It  is  seen  that  in  the  case  of  Washington,  Fig.  77,  that  I0  is  a  line 
sloping  downward  to  meet  h  sloping  upward  in  the  contact  point 
on  the  ordinate  Ia  =  1.528;  in  the  case  of  Bassour  I0  is  a  hori- 
zontal line  meeting  the  upward  sloping  h  in  the  contact  point 
1.680;  in  the  case  of  La  Quiaca  I0  is  an  upward  sloping  line  to 
meet  /i  in  the  contact  point  2.010.  In  each  case  the  contact 
point  is  on  the  ordinate  axis  to  which  corresponds  the  coefficient 
of  transmission  pw  =  1.000,  which  is  that  for  perfect  transmission. 
Whatever  may  be  the  physical  cause  of  the  sloping  of  the  I0 
lines,  below  or  above  the  horizontal,  the  contact  point  has 
eliminated  that  cause  from  the  system.  In  Case  III,  Wash- 
ington, it  is  necessary  to  depress  the  sloping  line  I0  into  a  hori- 
zontal position  Ia')  in  Case  II,  Bassour,  this  is  already  done;  in 
Case  I,  La  Quiaca,  the  sloping  line  must  be  raised  to  the  horizontal 
position  7a.  In  Case  III,  if  Ii/pw  =  I0,  then  Ii/pa  =  /a, 
where  pa  is  larger  than  pw;  in  Case  I,  if  h/pw  =  I0,  then 
I\/pa  =  fa,  where  pa  is  smaller  than  pw.  It  is  necessary  to 
determine  an  equation  for  each  station  depending  upon  the 
aqueous  vapor  pressure  e  which  will  convert  pw  into  the  required 
pa.  This  is  done  in  Section  2  of  Table  90.  Assume  plw  at  con- 
venient intervals  1.000,  0.980,  0.960,  .  .  .  ;  take  the  contact 
point  I10  from  the  diagram,  and  71!  from  the  mean  line  on  the 


396 


EXTENSION    OF    THERMODYNAMIC    COMPUTATIONS 


<N  <N  (N  >O  00  <N  «O 
i-i  rH  CO  <M'  >O  cd  "3 


I   IO  CO  O  CO  rH  TJH 

^   l>  1>  t>  CO  CO  *O 


CO  O5  C^l  1^  CO  rH 

O5  t^*  CO  C^Q  CO  Oi 
00  00  00  00  00  *>• 


<N  iO 

(N  rH 

O5  O5 


>O  O  Oi 

O5  00  CO 
00  00  00 


OO  O5  1 

rH  (N  (N  rH 


CO1>I>OOOOCO001>1>1>CO 


t—t^-C^i—  1 

l^-  O5  l>-  »O 


CO  CO  CO  t>*  CO  CO  CO  CO  CO  CO  CO 


1      1   I   I   I   I 


Ot>COOOTjHOCD<N 
O  rH  CO  TjJ  cd  00  CJ  i-i 


!>•  ^  cq  O5  CD  co  o 

SCO  O  CO  CO  O  !>• 
O5  O5  00  00  OO  t*- 


CO  ^^ 

O  O5  00  00  1> 


c 


oooooooo 
oooooooo 


rfi  <N  O  00  CO      •      •      • 
O^  O^  O^  00  00      •      • 


CDcOcOcOcococO 


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O^  O^  O5  O5  00 


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oooooooooooooooooooooo 

(N(M(M(M(N(M(M<N(N(N(N 


COMPUTATION    OF    Pa    AND    Ia 


397 


diagram;  then  compute  pa  =  y.    Plot  down  e  on  an  auxiliary 

lo 

ordinate  scale  for  the  same  axis  of  abscissas,  and  draw  the  mean 
e.  Now  it  is  evident  that  if  the  aqueous  vapor  effect  is  to  be  elimi- 
nated the  line  e  should  pass  through  the  origin  where  pw  =  1.000. 
In  order  to  do  this,  draw  a  line  through  the  origin  parallel  to  e, 
that  is  (e  +  2.4)  mm.  for  Washington,  (e  —  7.5)  mm.  for  Bassour, 
and  (e  +  5.0)  mm.  for  La  Quiaca.  Take  pa  —  pw  =  F  (e  +  B\ 
where  F  =  the  coefficient,  in  fact  the  ratio  of  the  ordinate  of  e 
to  the  ordinate  of  h  counted  from  the  horizontal  line  7a.  Hence, 
we  find  the  equation, 


where  B  is  the  vertical  ordinate  between  the  two  vapor-pressure 
lines.  In  type  II,  pa  =  pw  ;  in  type  III,  pa  =  pw  +  F  (e  +  B)  ; 
in  type  I,  pa  =  pw  —  F  (e  +  B).  At  each  station  there  is  to  be 
computed  such  a  value  of  the  coefficient  of  transmission  pa  as 
will  make  the  values  of  the  extrapolated  /i  fall  upon  the  hori- 
zontal line  /a,  instead  of  upon  the  sloping  line  I0.  If  they  do 
fall  upon  a  sloping  line,  it  follows  that  the  resulting  mean  values 
of  a  series,  winter  series,  for  example,  will  differ  radically  from  the 
summer  series,  because  the  means  will  pertain  to  different  groups 
of  pw.  Take  the  mean  values  of  each  half  of  the  groups  in 
Section  1: 


Station 

Pw 

/o 

Pw 

/o 

Difference 

Washington  
Bassour       

0.854 
0.869 

1.633 
1.676 

0.772 
0.763 

1.687 
1.674 

+  0.054  Type  III 
-0.002  Type    II 

La  Quiaca 

0  914 

1  893 

0  876 

1  819 

-0  074  Type      I 

This  is  evidently  one  cause  for  the  incessant  variations  that 
characterize  pyrheliometer  mean  values.  It  is  plain  that  many 
such  fluctuations  as  have  been  attributed  to  solar  action  are,  in 
fact,  due  to  the  imperfect  elimination  of  the  terrestrial  effects  of 
aqueous  vapor  and  dust  as  well  as  density  from  the  intensity 
of  radiation  at  the  station. 


398 


EXTENSION  OF  THERMODYNAMIC  COMPUTATIONS 


Summary  of  the  Correction  Equations 


Calories 


Washington  D.O.,  Height  34m. 


Ba88ourAlgeria,HeightllCOm.     ^  Quiaca Height 


2.000 


Type  III 


Type  II 


Type  I 


1.900 
1.800 


1.700 
1.600 
1.500 


1.400 
1.300 


1.200 
1.100 
1.000 


O  O  00 


e'=(e- 


7.5)  =  O 


pw.    1.000          0.900 


0.800 


1.000 


0.900 


0.800 


1.000          0.900 


FIG.  77.  —  Three  types  of  pyrheliometric  data 

Type  I. 

La  Quiaca,  height  3,465  meters,  pa  =  pw  -  0.0080  (e  +  5.0). 
Mt.  Wilson,     "      1,780        "      pa  =  pw  -  0.0020  (e  +  2.0). 
La   Confianza    (4,483),   Mt.    Whitney    (4,420),   Humahuaca 
(2,939),  Maimara  (2,384),  have  similar  equations,  but  the 
series  of  observations  is  too  short  to  determine  the  constants 


Type  II. 

.  Bassour,        height  1,160  meters,  pa  =  #«,. 
Mt.  Weather,    "         526       "       pa  =  pw. 
There  is  some  uncertainty  about  the  equation  for  Mt.  Weather 
on  account  of  the  indecisive  data  of  I0. 

Type  III. 

Jujuy,  height  1,302  meters,  pa  =  pw  +  0.0060  (e  -  6.0). 
Cordoba,  "  438  "  pa  =  pw  +  0.0087  (e  -  2.6). 
Pilar,  "  340  "  pa  =  pw  +  0.0087  (e  -  2.6). 

Potsdam,  "  89  "  pa  =  pw  +  0.0090  (e  +  0.0). 
Washington"  34  "  pa  =  pw  +  0.0100  (e  +  2.4). 

The  coefficient  F  diminishes  from  +  0.0100  at  Washington 


REDUCTIONS    TO    SEA    LEVEL 


399 


(34)  through  0.000  at  Bassour  (1,160),  to  -  0.0080  at  La  Quiaca 
(3,465);  the  amount  of  the  B  is  very  variable  with  the  local 
conditions  of  the  vapor-pressure  e  at  the  station. 

Apply  these  equations  to  the  values  of  /i  and  compute  7fl. 


TABLE  91 
WASHINGTON,  D.  C.,  STATION  EQUATION,  pa 


pw+  0.0100  (eX2A) 


No. 

Pw 

e 

e  +2.4 

A 

Pa 

* 

la 

Isea 

A/ 

7 

.884 

3.78 

6.2 

.062 

.944 

1.410 

1.497 

5 

- 

16 

.874 

3.00 

5.4 

.054 

.928 

1.402 

1.511 

*t 

Js 

20 

.865 

2.09 

4.5 

.045 

.910 

1.409 

1.548 

rH 

^ 

16 

.854 

2.60 

5.0 

.050 

.904 

1.392 

1.540 

*0> 

<fl 

22 

.846 

3.34 

5.7 

.057 

.903 

1.386 

1.535 

> 

B 

24 

.834 

4.06 

6.5 

.065 

.899 

1.410 

1.568 

"J 

fi 

17 

.824 

3.80 

6.2 

.062 

.886 

1.353 

1.527 

• 

'~ 

rt 

17 

.815 

3.69 

6.1 

.061 

.876 

1.418 

1.619 

ti 

§ 

10 

.804 

4.30 

6.7 

.067 

.871 

1.334 

1.532 

^ 

JS 

mean  1.542 

14 

.795 

5.02 

7.4 

.074 

.869 

1.349 

1.553 

*«-, 

** 

18 

.784 

5.10 

7.5 

.075 

.859 

.316 

1.532 

0) 

1 

8 

.773 

5.10 

7.5 

.075 

.848. 

.303 

1.537 

•— 

9 

.764 

5.19 

7.6 

.076 

.840 

.335 

1.590 

> 

d 
o 

11 

.756 

6.10 

8.5 

.085 

.841 

.272 

1.512 

bit) 

'§ 

7 

.743 

7.03 

9.4 

.094 

.837 

.260 

1.506 

g 

E 

5 

.736 

7.69 

10.1 

.101 

.837 

.262 

.508 

8 

5 

.714 

8.85 

11.3 

.113 

.827 

.252 

.514 

<u 

o 

3 

.703 

7.40 

9.8 

.098 

.801 

.170 

.461 

H 

H 

4 

.696 

6.06 

8.5 

.085 

.781 

.212 

.552 

mean  1.527 

.534 

1.525 

-0.009 

The  computed  values  of  Ia  from  pa,  in  place  of  I0  from  /?„,, 
plot  near  the  dotted  line  in  the  diagram.  The  mean  of  the  first 
nine  values  is  1.542,  and  of  the  last  ten  values  1.527,  showing 
that  the  variations  in  pw,  and  the  effect  of  the  vapor-pressure  e, 
are  quite  well  eliminated.  A  longer  series  would  probably  prove 
that  the  elimination  is  complete. 

Table  92  contains  the  summary  of  the  computed  Ia  for  nine 
stations  and  their  mean  values.  If  these  mean  values  of  Ia  are 
plotted,  for  the  height  z  on  the  axis  of  ordinates  and  calories 
on  the  axis  of  abscissas,  the  points  fall  nearly  on  a  straight  line, 
so  that  Ia  diminishes  in  proportion  to  the  height.  This  line  cuts 
the  sea  level  at  1.525  calories,  and  the  differences  which  corre- 


400 


EXTENSION     OF     THERMODYNAMIC     COMPUTATIONS 


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spond  to  the  station  height  are  readily  computed.  It  is  equally 
easy  to  determine  the  normal  value  of  Ia  at  any  station,  and 
several  examples  are  added  for  stations  having  only  short  series. 

The  Annual  Mean  Variations 

The  station  equations  can  be  applied  to  the  individual  pw  — 
groups,  or  to  individual  observations,  during  different  years, 
and  their  mean  sea-level  values  are  collected  in  Table  93;  also, 
the  annual  means  are  taken  for  all  stations  which  were  observed 
during  the  same  year.  We  have  such  means  extending  from 
1903  to  1914,  and  they  are  plotted  on  the  upper  curve  of  Fig.  78, 
giving  minima  in  1903,  1907,  1912,  1914.  The  1907  and  1912 
minima  are  already  well  known.  In  the  second  curve  of  Fig.  78 
is  reproduced  the  adopted  mean  meteorological  curve  from 
Bulletin  No.  1,  Oficina  Meteorologica  Argentina,  written  in 
1910,  which  contained  a  summary  of  data  till  1910,  and  a  forecast 
from  1910  to  1915.  It  should  be  noted  that  the  forecast  for  1911, 
1912,  1913,  was  very  well  verified,  as  it  has  actually  been  in  all 
other  of  the  Argentine  meteorological  data.  In  1914  the  mini- 
mum, which  the  forecast  placed  in  1915,  seems  to  have  come  in 
1914,  but  this  is  very  unexpected,  because  there  is  an  interval  of 
only  two  years  following  1912,  whereas  the  ordinary  periodic 
interval  is  3.75  years.  Another  forecast  is  added  with  maxi- 
mum in  1917  and  minimum  in  1919.  It  is  necessary  to  maintain 
suitable  solar  observations,  in  order  to  study  the  causes  of 
such  irregular  fluctuations  in  the  output  of  the  solar  radiation, 
and  several  solar  physics  observatories,  adapted  to  meteorolog- 
ical purposes,  are  indispensable  in  the  interests  of  long-range 
forecasts. 

The  observatories  at  Pilar  and  La  Quiaca,  Argentina,  are 
well  adapted  to  supplement  the  work  of  Washington  and  Mt. 
Wilson  in  the  United  States. 

The  General  Summary 

It  has  been  shown  that  the  mean  values  of  Ia  plot  along  a 
straight  line  with  a  given  slope.  On  Fig.  76  is  plotted  S  =  3.95 
the  solar  constant;  S  Ja  the  curve  of  the  absorbed  radiation  from 


402 


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403 


the  outer  limit  to  the  station  level,  or  it  may  be  taken  as  (S  — 
2  Ja)  =  By  plotting  S  Ja  from  the  ordinate  S;  the  density 
curve  p  is  plotted  with  the  abscissa  as  indicated;  the  pyrheliom- 
eter  curve  Ia  =  A  is  plotted  up  to  4,500  meters,  and  its  course 
follows  the  p  curve  so  closely  as  to  suggest  that  these  two  func- 
tions run  out  together,  that  is,  at  the  solar  constant  3,950  calories. 
If  Abbot's  ordinates  for  the  bolometer  are  filled  up  by  proper 
interpolations  for  the  several  intermediate  wave  lengths,  and 
the  sums  taken,  we  shall  have  good  relative  values  of  the  several 
bolometer  energy  areas. 

TABLE   94 
SUMMARY  OF  THE  BOLOMETER  DATA 


Bolometer  Data 

Abbot's 
Scale 
Sum. 

Calories 

Factor 
of 
Reduction 

Calories 
B 

Height 

z 

Black  body  6900°  A  
Abbot's  extrapolated  

80.30 
67.26 

3.876 

20.7 

3.876 
3.25 

23000 
6500 

Mt.  Whitney  

61  38 

2  97 

4420 

Mt.  Wilson 

59  55 

2  88 

1780 

Washington,  D.  C  

51.19 



2.47 

34 

At  the  solar  temperature  6900°  A,  for  black  body  radiation, 
the  value  at  the  outer  limit  of  the  earth's  atmosphere  is  3.876, 
and  corresponding  to  this  the  sum  of  Abbot's  ordinates  on  his 
arbitrary  scale  is  80.30.  The  factor  of  reduction  is  taken  20.7, 
and  the  calories  B  for  the  several  stations  at  the  height  z  follow. 
Plot  these  on  Fig.  76,  and  they  seem  to  belong  to  the  S  Ja  curve. 
We  conclude  that  the  pyrheliometer  data  approximate  the  p 
density  curve,  and  the  bolometer  data  represent  the  selective 
radiation  curve  S/a.  The  total  solar  radiation  now  divides 
itself  into  three  parts: 

(1)  Radiation  intensity  S  Ja  used  in  producing  temperature. .  1.46 

(2)  Radiation  energy  scattered  back  to  space 1.00 

(3)  Free  radiation  energy  penetrating  to  the  sea  level ,  .  1.52 

Total  or  solar  constant  energy  in  calories 3.98 

The  corresponding  temperature  would  be  6930°  A. 
The  7o  —  curve  of  the  first  method  has  been  changed  into 
the  Ia  -  curve  by  the  second  method  of  computation. 


404          EXTENSION    OF    THERMODYNAMIC   COMPUTATIONS 


GENERAL  REMARKS 

On  Fig.  76  are  drawn  the  typical  curves  from  the  data  of 
Chapter  VII:  Ja  the  selective  radiation  absorbed  in  each  1,000- 
meter  stratum,  and  2  Ja  the  total  amount  absorbed  down  to 
the  given  level;  J0  the  black  body  radiation  in  each  1,000-meter 
stratum/ anpl  S/0  the  total  amount  down  to  the  given  level; 
/i  the  computed  radiation  for  the  zenith  sun  on  each  level,  I0 
the  computed  Ii/pw,  and  Ia  =  Ii/pa;  the  density  curve  p;  the 
solar  constant  S  =  3.950,  and  Abbot's  solar  constant  £  =  1.950. 

(1)  Abbot  multiplies  I0  by  a  small  bolometer  factor,  and  adds 
a  small  function  of  e  to  produce  E  =  1.950  on  each  level. 
Bigelow's  curve  Ia,  derived  from  the  contact  points  for  pw  = 
1.000  at  perfect  transmission,  crosses  I0  at  an  angle,  because 
Abbot's  mean  values  of  I0  are  too  large  at  low-level  stations,  and 
too  small  at  high-level  stations,  both  systems  agreeing  for  pw  = 
pa  at  about  1,200  meters,  the  level  of  the  cumulus  cloud  base. 
The  pyrheliometer  curve  A,  following  closely  upon  the  density 
curve  p,  will  run  out  at  about  3.950  colories,  in  conformity  with 
the  S/0  =  3.950  calories. 

(2)  The  bolometer  ordinates  indicate  an  amount  of  energy 
well  represented  by  the  curve  S/0  or  B,  so  that  Abbot's  small 
bolometer  factor  should  be  greatly  increased,  up  to  2.00  in  the 
higher  levels.    The  theory  which  led  to  his  small  bolometer 
factor,  namely,  that  the  pyrheliometer  registers  the  same  amount 
of  heat  as  that  indicated  by  the  bolometer,  when  the  small 
corrections  have  been   applied,   is   erroneous.     The   curve  I0 
does  not  represent  the  true  pyrheliometer  intensities,  which  are 
7fl,  nor  does  it  take  any  account  of  the  energy  scattered  in  the 
area  between  A  and  B,  nor  of  the  heat  absorbed  in  making  the 
temperatures  T  in  the  area  between  B  and  S. 

(3)t  The  pyrheliometer  does  not  directly  measure  the  solar 
constant  at  low  levels,  and  the  Bouguer  formula  is  incapable 
of  making  the  complete  extrapolations;  the  bolometer  does  not 
measure  the  solar  constant  except  by  an  approximate  inference; 
the  thermodynamics  of  the  atmosphere  affords  data  for  com- 


GENERAL   REMARKS  405 

puting  S/a  and  S/0  in  all  levels,  and  the  final  sum  for  S/0  at 
the  sea  level  is  the  solar  constant,  about  3.950  calories. 

(4)  The  thermodynamic  computations  are  in  accord  with  this 
interpretation  of  the  pyrheliometric  data  and  the  bolometric 
data,  hi  requiring  about  3.950  calories  to  do  the  work  actually 
measured  in  the  atmosphere,  and  it  would  be  impossible  to 
maintain  the  existing  temperature,  pressure,  and  density  distri- 
butions by  expending  only  1.950  calories.  The  pyrheliometer  is 
useful  in  determining  the  relative  annual  variations  of  the  solar 
radiation  at  any  station;  the  bolometer  is  necessary  in  studying 
the  relative  absorptions  of  different  wave  lengths  in  the  atmos- 
pheres of  the  sun  and  of  the  earth;  but  the  thermodynamic 
data  are  required  for  the  mutual  interpretation  of  these  two 
types  of  data. 

This  subject  is  treated  at  greater  length  in  Bulletin  No.  4, 
Oficina  Meteorologica  Argentina,  1914. 


406  EXTENSION    OF    THERMODYNAMIC    COMPUTATIONS 


The  Constants  and  Coefficients  of  Dry  Air  in  the  Kinetic  Theory 
of  Gases  throughout  the  Atmosphere 

It  facilitates  the  discussion  of  the  problems  of  the  upper 
layers  of  the  earth's  atmosphere  to  have  the  data  of  the  Kinetic 
Theory  of  Gases  for  Dry  Air  in  a  form  for  reference.  Such  a 
table  of  constants  and  coefficients  for  the  (M.  K.  S.)  and  the 
(C.  G.  S.)  systems  is  given  in  Table  95,  and  the  results  of  their  ap- 
plication from  the  bottom  to  the  top  of  the  earth's  atmosphere 
as  computed  in  Tables  96,  97.  The  fundamental  data  for  N  = 
6.062  X  1023,e  =  4.774  X  10 ~10,  conform  to  the  summary  of  con- 
stants by  R.  A.  Millikan,  Physical  Review,  August,  1913,  the 
values  for  n,  EQ,  e,  k,  h,  cz,  differing  a  little  in  consequence  of 
slight  variations  in  the  adopted  meteorological  data  for  P,  p,  R. 
It  is  interesting  to  note  that  these  fundamental  physical  quan- 
tities, and  many  others,  can  be  computed,  not  only  at  the  surface 
in  standard  conditions,  but  throughout  the  atmosphere.  Sim- 
ilarly, the  sun's  data  can  be  computed  provided  P,  T,  p,  R}  can 
be  secured  at  the  required  points  in  the  solar  atmosphere.  The 
formulas  of  Table  95  indicate  the  method  of  the  computations, 
and  numerous  subordinate  formulas  can  be  derived  from  them. 
The  transformation  factors  between  the  (M.  K.  S.)  and  the 
(C.  G.  S.)  systems  have  been  checked  throughout  this  group  of 
Formulas.  In  making  transformations  from  (M.  K.  S.)  to 
(C.  G.  S.)  M  =  103,  L  =  102;  and  from  (C.  G.  S.)  to 
(M.  K.  S.)  M  =  10~3,  L  =  10-2. 

In  Table  96,  the  kinetic  energy  per  unit  volume  Hy  and  U 
the  inner  energy  per  unit  volume,  decrease  in  a  curve  similar 
to  those  of  P  and  K  to  vanishing  values;  q,  the  mean  square 
velocity,  and  T,  the  arithmetical  mean  velocity,  diminish  till 
they  disappear  at  the  outer  limit;  N,  the  number  of  molecules 
per  kilogram-molecule,  and  n,  the  number  of  molecules  per 
cubic  meter,  diminish  slowly,  but  do  not  vanish.  If  the  term  N 
ought  to  remain  constant,  this  diminution  may  go  back  to  some 
additional  physical  forces  not  included  in  these  simple  formulas. 
Compare  Fig.  70  for  typical  curves  of  the  Dynamic,  Thermo- 


THE  CONSTANTS  AND  COEFFICIENTS  FOR  DRY    AIR          407 

dynamic,  and  the  Kinetic  Theory  data,  throughout  the  atmos- 
phere. 

A  second  more  refined  computation,  with  improved  tempera- 
tures, has  been  undertaken.  The  data  of  Table  97  should  be 
substituted  for  those  of  Table  96  above  40,000  meters,  and 
the  following  remarks  are  based  upon  it.  lmax,  Maxwell's  free 
path,  increases  from  the  surface  upward,  rapidly  above  45,000 
meters ;  v,  the  number  of  collisions  per  second,  diminishes  with 
the  height.  The  supposed  height  of  the  atmosphere,  as  derived 
from  meteors  at  100  kilometers,  finds  complete  verification  in 
these  computations.  mHj  the  mass  of  the  hydrogen  atom,  as 
computed,  shows  a  small  increase  in  value,  probably  due  to 
some  defects  in  our  adopted  simple  formulas.  Similarly,  e_,  the 
elementary  negative  electric  charge,  assumed  in  this  formula 
to  hold  a  constant  ratio  to  mH,  3.4554  X  10~13  (M.  K.  S.), 
follows  the  fortunes  of  mB. 


408 


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414  EXTENSION    OF    THERMODYNAMIC    COMPUTATIONS 


Summary  of  the  Dimensions  with  Special  Reference  to  the 
Equivalents 

In  transformations  from   the   (M.  K.  5.)   to   the   (C.  G.  S.) 
systems,  the  following  dimensions  should  be  observed: 

Mass  =  M=103.  Length  =  L  =  102.  Time=r=l  second. 
Velocity  =  L  T~l.  Momentum  =  M.L  T~l.  Acceleration  = 

LT.~l  T.~\ 
Force  =  M.L  T  ~l  T~l  =  M  L  T~2.      Impulse  =  M.L  T~l. 

Force  per  unit  mass  or  force  of  acceleration      .  =  L  T~2 
Force  per  unit  volume  =  L  T~2.  M  L~*      .      .  =  M  L~2  T~2 

I  Specific  Volume  =  Volume  per  unit  mass    .      .  =  M~l  L3 
I  Specific  Density  =  Mass  per  unit  volume     .     .  =  M  L~3 

Work  =  force  X  length  =  M  L  T~2.  L  .  .  =  M  L2  T~2 
Kinetic  Energy  =  Mass  times  velocity  squared  =  M  L2  T~2 
Increase  of  kinetic  or  potential  energy  =  work 

done        .....     ........  =  ML2T~2 

I  Kinetic  Energy  per  unit  mass     ......  =  L2  T~2 

|  Gravity  potential  =  work  per  unit  mass       .     .  =  L2  T~2 

Kinetic  Energy  per  unit  volume  =  L2  T~2  .  M  L~*  =  M  L~l  T~2 
Pressure  =  force  per  unit  area  =M  L  T~2.  L~~2  =  M  L~l  T~2 
Pressure  =  work  per  unit  volume  =ML2  7^2.  IT3  =M  L~l  T~2 


In  the  equations  of  Condition,  we  have  for  T  =  1, 
p  _  p 

,  g  (Zl  -z0)=  -  -%  (qi*  -  g02)  -  (Ql  -  Qo),  s 

Pio 

L.L  =  M  L~\  M~1L3  =  L2  =  L2  =  104  =  10,000. 


i=L2.M  L-*= 


Pressure  has  two  definitions:  (1)  force  per  unit  area,  as  Kilo- 
grams per  square  meter,  or  grams  per  square  centimeter;  (2)  work 
per  unit  volume,  as  free  heat  per  unit  volume,  hydrostatic  pres- 


SUMMARY    OF    DIMENSIONS  415 

sure,  inner  energy  per  unit  volume,  radiation  energy,  and  these 
all  have  the  same  dimension,  M  L~l.  Since  the  Erg  and  the 
Joule  are  units  of  work  they  must  refer  to  the  unit  volume  and 
not  to  the  unit  area. 


Hence  =  MLi  T~\  L~*  =  M  ZT1  T~*  =  103X  1(T2=  10, 


While  -~  =  ML*  T~\  L~*  =  M  T~*  =  103  =  1,000 
is  distinctly  erroneous. 

Summary  of  Dimensions 

In   the   conversion  from  (M.  K.  S.)  mechanical  units  into 
gr.  cal./cm.2  minute,  the  correct  factor  becomes: 

(M.K.S.)  X  10  X  60/4.1851  X  107  =  0.000014336 

On  the  other  hand,  should  the  area  L?  be  placed  in  the  denomi- 
nator under  Joule,  instead  of  the  volume  Z,3,  the  factor  (M.  K.  S.) 
X  1000  X  60/4.1851  X  107  =  0.0014336  would  be  incorrect,  in- 

asmuch as  the  —  -  —  -  is  the  work  energy  per  unit  volume  and 

not  per  unit  area. 

The  equations  of  condition  assume  two  typical  forms: 

(a)  -  (Pi  -  Po)  -£  =  g  (zi  -  z0)  fto-j, 

M  L~l  =  L.L.  M  L~*  =  M  L~l  =  10 
,M        (Pi  -  Po)  60  60 

(b)  •    -^r"^  =  g(Si"2o)^ 

M  L~l.  M~l  D  =  L  .  L  =  D  =  104 
For  the  equation  (a),  from  (M.  K.  S.)  Mech.  through  (C.  G.  S.) 

Mech.,  (M.  K.  S.)  X  10  X  60/4.1851  X  107  =  0.000014336 
For  the  equation  (a),  from  (M.  K.  S.)  Mech.  through  Kil.  Cal./ 

m.2  min.,  (M.  K.  S.)  X  60/4.1851  X  103  X  103  =  0.000014336 
For  the  equation  (b),  from  (M.  K.  S.)  Mech.  through  (C.  G.  S.) 

Mech.,  (M.  K.  S.)  X  104  X  60/4.1851  X  1010  =  0.000014336 
For  the  equation  (b),  from  (M  .  K.  S.)  Mech.  through  Kil.  Cal  ./M2 

min.,    (M.  K.  S.)  X  60/4.1851  X  103  X  103  =  0.000014336 


416  EXTENSION    OF    THERMODYNAMIC    COMPUTATIONS 

TABLE  98 — SUMMARY  OF  DIMENSIONS 


Formulas 

Dimensions 

M.  K.  S. 

C.  G.  S. 

Work 

,_kilog.m2 

gr.  cm.2 

P  —  pR  T 

ML        1         M 

kilog.       Joule 

sec.2 
gr.            erg 

p  (m)  =  mol.  wt. 
U. 

T2         L2     L  T2 

~L? 
L3 

m.  sec.2        m3 
kilog. 
m3 

m* 

cm.  sec.2      cm3. 
gr. 
cm.3 

cm.3 

(M)  =  M  (m) 
R  Cp  Cu 

M 
L* 

kilog. 

gr. 
cm.2 

K  =  R(m) 

T^deg. 
L2(m) 

sec.  Meg. 
m2(m) 

sec.2  deg. 
cm.2(wi) 

A  =  mech.  eq.  H 

T2deg. 
LML 

sec.2  deg. 

4  13*51  V  1fV    kilo&-  m* 

sec.2  deg. 

4  IS'JI  V  1010  /oor  2 

Watt 

T^cal 

(M) 

sec.2 
kilog.               Joule 

1  Watt.             107  erg 

L2T 

N       1 

r3  deg.« 

1 

sec.3  deg.4     m2  sec.  deg.4 
1 

cm.2  deg.4    cm.2  sec.  deg.4 
1 

n 

M 

I 

mass  hydrogen  atom 
1 

mass  hydrogen  atom 

1 

LML-i     (m)U      (m)L2 

m3  (mol.  wt.) 
(mol.  wt.)  m2 

cm.3  (mol.  wt.) 

Pv—  RT 

T2           M            T2 
M  L~i    L3      L2 

sec.2 
m2 

sec.2 
cm2. 

2      3P0 

N 

T2        M~  T2 

sec.2 

,         ,        m* 

M-mol.     ; 
sec.2 

m2 

sec.2 

M-mol.  —  '— 
sec.2 

cm.2 

2  •  V- 
77        3    ^ 

T2 

sec.2 

sec.2 

2    n 

(M)L2 

m     sec.2 

sec.2 
(M)  cm.2 

t             **• 

T2  deg. 

sec.2  deg. 
kilog.  m2     Joule 

sec.2  deg. 
gr.  cm.2       erg. 

AT 

L 

T2  deg. 
ML2     ML* 

sec.2  deg.       deg. 
kilog.  m2 

sec.2  deg.       deg. 
gr.  cm.2 

Ci  =  Sirch 

f                  y2 

ML? 

_„,            J  ouie  sec. 
sec. 

kilog.  m3 

sec. 
gr.  cm.3 

C2=ch/k 

Ldeg. 
."z.  r~2  ,     s/  /•—  i  y—  2 

sec.2           Joule  meter 
meter  deg. 
kilog.         Joule 

sec.2 
cm.  deg. 
gr.             erg 

H      2P 

TJ         C     n  T 

•pi                      M  L.       1     - 

L2       M  ,              M 

met.  sec.2         m3 
kilog.        Joule 

cm.  sec.2        cm.3 
gr.               erg 

C 
g 

a  =  

T2  deg.'  L3  deg>  ~  L  T2 
M  L\      1                M 

m  sec.2         m3 
w/sec. 
w/sec.2 
kilog.                 Joule 

cm.  sec.2        cm.s 
cm./sec. 
cm./sec.2 
gr.                       erg 

a  c 

T2  L4deg.4     LT2deg.4 
M          L         M 

m.  sec.2  deg.4      m3  deg.4 
kilog.                Joule 

cm.  sec.2  deg.4       cm.3  deg.4 
gr.                       erg 

4 

cr 

L  r2  deg.4'  T     r3  deg.4 
M       T2  cal.      M  cal. 

sec.3  deg.4      m.2  sec.  deg.4 
kilog.  cal. 

sec.3  deg.4       cm.2  sec.  deg.4 
gr.  cal. 

^A      A 
<r 

r'deg.4       L2       "  L2rdeg4 

m2  sec.  deg.4 
kilog.                 Joule 

cm.2  sec.  deg.4 
gr.                       ere; 

^ 

T3  deg.4 

sec.3  deg.4       m2  sec.  deg.4 

sec.3  deg.4      cm.2  sec.  deg.4 

GENERAL    PROBLEMS  417 

It  should  be  noted  that  a  X  j,  =  cr,  so  that  a  is  the  work 

energy  per  unit  volume.  When  this  gains  the  velocity  L  Ti, 
it  becomes  cr  or  radiation  through  the  unit  area.  The  equation 
(b),  page  414,  KIQ  =  cTa  gives  c  of  the  same  dimension  as  a, 
and  it  must  be  multiplied  by  a  velocity  to  become  comparable 
with  o-  in  the  Stefan  Law.  The  computations  for  the  radiation 
coefficients  conform  to  these  dimensions. 

Similar  computations  have  been  successfully  made  for 
hydrogen  and  calcium  vapor  in  the  atmosphere  of  the  sun,  and 
the  results  are  in  conformity  with  the  indications  of  the  well- 
known  observations.  This  knowledge  of  P.p.R.T.  in  the 
superficial  layers  of  the  sun's  envelope  is  of  great  value  in  in- 
terpreting the  lines  of  the  visible  and  radiation  spectra.  Indeed, 
the  thermodynamic  data  are  so  complex  that  it  will  be  im- 
practical to  determine  them  by  direct  observations.  The  change 
of  R  =  constant  to  R  =  variable,  that  is,  the  transition  from 
adiabatic  to  non-adiabatic  conditions,  converts  the  so-called 
constants  k,  h,  c,  c2}  a,  a  into  a  series  of  coefficients  which  change 
from  level  to  level.  It  follows  that  the  laws  of  radiation  for 
each  chemical  element  are  much  more  complicated  than  is 
implied  in  the  use  of  the  Stefan  or  Wein-Planck  Laws  with 
simple  constants.  This  subject  is  of  such  far-reaching  im- 
portance that  it  is  reserved  for  further  study  and  research. 

General  Problems  in  Atmospheric  and  Solar  Physics 

The  purpose  of  the  development  of  non-adiabatic  meteorology 
has  been  to  discover  a  method  of  computations  applicable  to  all 
atmospheres,  as  those  of  the  earth  and  the  sun.  The  process 
consists  in  fixing  an  initial  P0,  /o0,  Rot  T0  on  some  level  z0j  and  then, 
by  assuming  7\  on  the  level  2i,  determine  the  corresponding  PI, 
Pi,  TI  on  that  level,  all  to  be  checked  by  the  gravity  equation. 
From  these  values  of  T  the  entire  system  can  be  obtained.  In 
the  case  of  the  sun  the  initial  data  can  be  fixed  approximately 
as  follows: 

If  the  Boyle- Gay  Lussac  Law  in  the  atmosphere  of  the  earth 


418 


GENERAL    PROBLEMS 


is,  P  U  =  R  T,  this  can  be  converted  to  the  sun's  atmosphere 
by  multiplying  each  term  with  28.028  =  G/g  =  T,  the  ratio 
of  the  surface  gravity  on  the  sun  to  that  on  the  earth,  so  that, 
PT  X  UT  =  RT  X  Tr.  The  corresponding  factor  for  all  other 
terms  follows  very  readily,  and  the  entire  thermodynamic 
system  can  be  developed  for  each  element  by  the  method  of 
trials.  The  check  is  found  in  the  gravity  equation. 


g 


p  _  p 

-  ZQ)  =  -        l  ~  (C  pa-  C  #10)  (Ta  ~  T0). 


The  analogy  between  the  physical  condition  in  the  atmosphere 
of  the  sun  just  above  the  photosphere,  and  in  that  of  the  earth 
between  the  levels  37,000  and  44,000  meters,  is  very  interesting. 
Compare  Fig.  79.  In  both  there  is  apparently  a  very  rapid 
change  of  temperature  with  the  height,  and  a  transition  from  ex- 
cessively rarefied  media  to  the  tru  egaseous  condition.  In  the 
earth's  atmosphere  there  is  passage  through  such  conditions  as  : 


z  in  meters 

37,000 

44,000 

T  (absolute) 
P 
B  (mercury  in  M  M) 

211° 
437 
3.3 

124° 
98 
0.7 

M.  K.  S.  Sys- 
tem of   units 

P 

0.025 

0.0087 

R 

82.77 

91.49 

The  physical  transformations  in  this  transition  at  44,000 
meters  correspond  with  those  sensitive  states  in  the  Geissler 
Tubes  at  which  the  electric  and  electro-magnetic  phenomena 
are  especially  active.  From  this  it  is  inferred  that  the  origin  of 
the  auroral  atmospheric  electricity  consists  in  the  transformation 
of  a  portion  of  the  incoming  solar  radiation  of  short  wave  lengths 
into  ions,  or  free  electric  charges,  and  that  these  pursue  their 
paths  to  the  polar  regions,  as  indicated  in  Fig.  69,  generating 
the  corresponding  magnetic  deflection  vectors.  In  Table  95  the 

mass  of  the  hydrogen  atom  is  mH  =  -r  =  1.6496  X  10~27,  and 


GENERAL  PROBLEMS  419 

if  this  is  a  constant,  N  =  6.062  X  1026  should,  also,  be  a  con- 
stant throughout  the  atmosphere.  But  by  Tables  96,  97,  N 
diminishes  with  the  height,  especially  above  40  km. 

z  =  90,000  N  =  1.665  X  1020 

80,000  1.064  X  1024 

70,000  7.023  X  1024 

60,000  2.216  X  1026 

50,000  5.364  X  1025 

40,000  1.092  X  1026 

30,000  1.730  X  1026 

20,000  2.671  X  1026 

10,000  4.161  X  1026 

000  No  =  6.147  X  1026 

The  differences  between  No  and  N  may  be  taken  as  the 
number  of  transformed  ions  on  the  several  levels,  to  be  used  in 
generating  the  auroral  electric  currents.  Furthermore,  the  region 
above  44,000  meters  in  the  earth's  atmosphere,  with  its  excessively 
rarefied  media,  in  electric  sensitiveness  under  the  impact  of  the  in- 
coming solar  radiation  of  short  wave  lengths,  is  distinctly  coronal  in 
its  physical  conditions,  and  it  corresponds  with  the  deep  corona 
that  exists  above  the  sun's  chromosphere.  The  rapid  changes  of 
temperature  near  the  sun's  photosphere  produce  a  layer  which 
is  the  source  of  great  radiation  energy  outward,  while  the  rapid 
temperature  increase  in  the  levels  44  to  37  km.  of  the  earth's 
atmosphere  is  the  evidence  of  an  important  absorption  of  radia- 
tion energy.  Hence,  the  sharpness  of  the  sun's  visible  disk  is  a 
phenomenon  depending  upon  the  thermodynamic  conditions 
existing  between  the  rapid  decrease  of  temperature  and  the  in- 
crease of  radiation  outward  to  space.  The  coronal  and  the  pho- 
tospheric  regions  in  the  atmospheres  of  the  sun  and  the  earth 
are,  therefore,  the  physical  counterparts  of  each  other. 

The  laws  of  the  absorption  of  the  incoming  radiation  in  the 
earth's  atmosphere  need  further  study.  In  place  of  interpreting 
the  bolometer  data  in  terms  of  the  pyrheliometer  data,  and 
ascribing  the  great  depletion  of  4.00  gr.  cal./cm.2  min.  down  to 
only  2.00  calories,  as  the  result  of  a  very  imperfect  action  of  the 
sun  as  a  black  body  radiator,  the  result  of  the  examination  of 
the  pyrheliometer  and  thermodynamic  data  seems  to  indicate 


420  GENERAL  PROBLEMS. 

that  the  pyrheliometer  data  should  be  interpreted  in  terms  of 
the  data  of  the  bolometer,  so  that  the  "Solar  constant"  is 
about  4.00  calories,  with  the  sun's  effective  radiation  tempera- 
ture 6950°  A.  In  the  Bouguer  formula,  the  coefficients  of  trans- 
mission pw  as  directly  observed,  and  pa  as  computed,  have  an 
interesting  relation  to  the  wave  lengths  in  the  spectrum.  Abbot's 
values  of  pw  at  different  wave  lengths  were  grouped  together, 
at  Mt.  Wilson  and  at  Washington,  and  diagrams  of  the  mean 
values  of  pw,  as  0.936,  0.925,  0.915  .  .  .  were  plotted  on  the 
abscissas,  0.20  M,  0.30  ju  ...  1.60  M.  If  the  values  of  pw  and 
pa  are  plotted  on  these  curves,  they  intersect  at  about  wave 
length  0.57  M.  If  0.57  M  on  pw  =  1.00  is  taken  as  the  maximum 
ordinate  on  the  energy  curve,  this  corresponds  with  about 
5000°  and  1.07  gr.  cal./cm.2  min.  at  the  distance  of  the  earth. 
Now,  this  is  the  average  direct  reading  of  the  pyrheliometer  at 
sea  level,  so  that  we  infer  that  not  only  the  aqueous  vapor  and 
dust,  but  also  the  density  of  the  atmosphere,  must  be  removed  in 
discussing  the  coefficients  of  transmission.  This  conforms  to  the 
result  shown  in  Fig.  76,  where  the  second  method  of  computa- 
tion identifies  the  depletion  of  Ia  =  I\/pa  with  the  density 
P  of  the  atmosphere.  This  research  is  not  yet  ready  for 
final  statement.  It  is  very  desirable  that  the  excellent  condi- 
tions prevailing  at  La  Quiaca,  at  the  elevation  3465  meters 
(11,037  feet)  on  the  Bolivian  Plateau,  should  be  utilized  for  a 
solar  physics  observatory,  equipped  with  a  bolometer,  spectro- 
heliograph,  and  direct-vision  spectroscope  for  prominences,  to 
be  operated  in  cooperation  with  Mt.  Wilson  and  Washington, 
D.  C.  The  conditions  of  living  in  La  Quiaca  are  comparatively 
easy  for  observers,  and  there  are  complete  railroad  facilities  for 
transportation  from  Buenos  Aires.  La  Confianza,  at  4483  me- 
ters (14,070  feet),  can,  also,  be  occupied  the  year  around,  as  the 
village  of  San  Vincente  and  the  neighboring  mining  camp  pro- 
vide the  necessary  accommodations.  It  is  important  that  the 
bolometric  data  should  be  obtained  at  these  high  levels,  and 
there  is  an  excellent  opportunity  for  an  expedition  to  make  the 
necessary  observations  at  those  stations. 

An  inspection  of  the  data  of  Chapter  VII  suggests  that  there 


GENERAL   PROBLEMS  421 

are  a  very  large  number  of  problems  in  general  physics  that  can 
be  studied  to  advantage  with  the  data  which  have  been  already 
developed.  The  laboratory  is  quite  incapable  of  furnishing  the 
fundamental  relations  that  are  shown  to  exist  above  the  45,000- 
meter  level,  where  the  pressure  is  a  small  fraction  of  one  milli- 
meter of  mercury.  Fortunately,  the  balloon  ascensions  to  30,000 
meters  cover  a  very  interesting  region,  the  so-called  "  isothermal 
layer,"  and  they  provide  quite  accurate  observations  of  the 
temperature.  The  most  important  outstanding  problem  is  to 
find  a  thermodynamic  function  for  the  kinetic  energy  of  circula- 
tion, —  ]4  (<?i2—  <?o2),  in  order  to  separate  it  from  the  free  heat 

-  (ft  -  90- 

It  is  hardly  possible,  in  opening  up  so  much  new  research 
material  in  the  physics  of  the  atmosphere,  to  have  escaped 
imperfections  and  even  errors,  but  it  is  thought  desirable  to  indi- 
cate to  meteorologists  and  astro-physicists  some  of  the  possible 
channels  of  investigation  that  appear  to  be  accessible  to  such 
studies  as  are  here  illustrated. 


INDEX 


Adiabatic  atmosphere,  V.  Bjerknes'  system,  51 

Margules'  system,  iii 

Adiabatic  and  non-adiabatic  systems,  differences,  51 
Angular  velocities,  cylindrical,  140,  142 

earth's,  160 

polar,  140,  142 

rectangular,  140,  142 
Aqueous  vapor  in  the  atmosphere,  grams  per  cubic  meter,  342 

grams  per  kilogram,  343 

Atlantic  Ocean,  meteorological  observations,  56 
Atmosphere,  homogeneous,  2 
Avogadro's  Law,  number  of  atoms,  28 
Azimuth,  coordinates,  7 

B 

Balloon  ascensions,  results,  115 

Huron,  So.  Dak.,  Sept.  1,  1910,  365,  377,  381,  410 

Twenty-one,  384 

Uccle,  June  9,  Sept.  13,  Nov.  9,  1911,  380,  385,  412 
Barometer  corrections,  for  local  effects,  45 

for  mean  column  temperature,  43 

for  plateau  temperature,  45 

for  removal  of  station,  42 

for  standard  gravity,  42 

for  standard  temperature,  41 

for  vapor  pressure,  45 

reduction  tables,  46 

to  normal  instrument,  42 
Barometric  formulas,  general,  39 
formulas,  23 

pressure,  annual  variations,  338 

transformations,  38 

V.  Bjerknes'  paper  on  Dynamic  Meteorology,  133 
Bolometer,  absorbed  energy  of,  393 
energy  spectrum,  278,  283 
instruments,  277 
observed  ordinates,  284 
Bouguer  formula,  application  in  atmosphere,  270 

indeterminate  development,  267 

remarks,  364 

423 


424  INDEX 

Boyle-Gay  Lussac  law,  P  =P  R  T,  1,  28 
Boyles'  law,  Mariotte,  28 


Calories,  large  and  small  units,  15 
Carnot's  function,  law,  71 
cyclic  process,  72,  73 

examples,  77 

Circulation,  direction,  and  velocity  on  the  earth,  190 
general  in  the  atmosphere,  189 
transformation  of  energy  in 

Case     I,  change  of  position  of  two  layers,  180 
Case    II,  adiabatic  and  non-adiabatic  mixture,  184 
Case  III,  overturn  of  strata,  185 

Case  IV,  masses  of  different  temperatures  on  the  same  level,  187 
Case    V,  discontinuous  temperatures,  188 
work  of,  153 

Clausius'  law  of  kinetic  energy,  28 
Coefficient  of  Conductivity,  discrepancy  of  results,  304 
examples,  305 
formulas,  295 

(c),  in  formulas  of  the  planetary  circulation,  126 
Components,  analytic  construction,  229 
general  and  local,  220 
graphic  construction,  232 
Concentration,  in  mixtures,  30 
Consecutive  means,  338 
Constants,  of  dynamic  meteorology,  9 
of  static  meteorology,  9 
of  thermodynamic  meteorology,  10 
three  systems,  13 

(M.  K.  S.)  and  (C.  G.  S.),  406,  408 

Continuity,  equation  of  in  rectangular,  cylindrical,  and  polar  coordinates,  145 
Convection,  diurnal  in  lower  strata,  98 

semi-diurnal  waves,  99 
Co-ordinate  accelerations,  136 

cylindrical  on  the  rotating  earth,  148 
forces,  137 

polar  on  the  rotating  earth,  149 
velocities,  136 
Co-ordinates,  rectangular,  cylindrical,  polar,  135 

systems  of,  8 

Coronas,  solar  and  terrestrial,  419 
Current  function,  in  angular  velocity,  171 

in  the  total  pressure,  171 
Cyclone,  example  of  the  evaluation  of  the  several  terms,  254 


INDEX  425 


Dal  ton's  law,  of  partial  pressures,  28 
Data,  Thermodynamic,  389,  393 
Deflecting  force  of  the  earth's  rotation,  178 
Density,  in  mixtures,  20 

in  the  isothermal  region,  96 

in  the  planetary  circulation,  118 

of  the  atmosphere,  13 

use  of  erroneous,  372 

variations  of,  17 

Differential,  total  of  -j-f  146 

Dimensions,  Conversion  Factors,  376 

Summary  of  equivalent,  414 
Diurnal  convection,  five  types  in  the  atmosphere,  312 

examples,  313 

magnetic  vectors,  compared  with  convection  vectors,  328 
variations,  electric  dissipation,  321 
magnetic  field,  323 
meteorological  elements,  318 
Dumb-bell-shaped  vortex,  equations  of,  177 
formulas  collected,  250 
tangential  angle  i,  251 
Dyne,  unit  of  force,  14 


Eastward  drift,  227 

Ebert  ion  counter,  formulas,  299 

example,  299,  301 

velocity  formulas,  300 
Electric  potential  of  the  atmosphere,  formulas,  309 

examples,  310 

Elster  and  Geitel  dissipation  apparatus,  formulas,  297 
Entropy,  (S\  —  So),  in  mixtures,  30 
in  the  isothermal  region,  93 
in  the  planetary  circulation,  124 
Ephemeris,  solar  magnetic,  334 
Equations,  general,  6 

of  motion,  cylindrical,    147 

in  cylindrical  vortices,  168 

in  Ferrel's  local  cyclone,  164 

in  the  German  local  cyclone,  166 

in  thermal  energy,  151 

First  form,   147 

Second  form,  149,  150 

Third  form,    152 

Fourth  form,  161 
Erg,  unit  of  work,  14 


426  INDEX 

Evaluation,  of  -jg,  159 

of  —  f  —  in  terms  of  temperature,  154 

for  standard  density,  157 

for  the  non-adiabatic  atmosphere,  156 

through  the  Boyle-Gay  Lussac  Law,  156 

183 


Evaporation  of  water,  laws,  343 
examples,  344,  345 
factors  for  pan  effect,  347 
formulas,  346 
from  soil  and  plants,  348 
Expansion  heat,  coefficient  of,  69 
Exponent  (a),  in  the  planetary  circulation,  126 

F 

Forces,  of  expansion  and  contraction,  137 
of  inertia,  137 
of  rotation,  139 

Formula  (172),  differentiation  of,  58 

Free  energy,  of  thermodynamic  potential  at  constant  volume  and  pressure,  67 
heat,  (Qi  —  Qo),  calories  as  unit  of,  15 
applications  of  formulas,  80 
evaluation  in  balloon  ascensions,  61 
Friction,  force  of,  179 
Funnel-shaped  vortex,  equations  of,  175 

G 

Gas  coefficient  R,  as  a  constant,  1 

as  a  variable,  16 

in  a  mixture,  29 

in  the  planetary  circulation,  118 
Gerdien  apparatus,  for  number  and  velocity  of  ions;  formulas,  302 

examples,  303 

Gold's  paper,  on  radiation  and  absorption,  134 
Gradient  ratio  n,  5 

variable  values  of,  57 
Gradients,  adiabatic  and  non-adiabatic,  4 
Gravitation,  force  of,  in  altitude,  19 

in  latitude,  18 

in  metric  and  English  measures,  20 
Gravity  potential,  67 

H 

Heat,  evaluation  through  the  entropy,  69 
in  anticyclones,  109 
in  cyclones,  108 


INDEX  427 


Heat  energy,  application  of  formulas,  77 

formulas,  66 

in  the  isothermal  region,  92-96 
inner  energy  of,  67,  69 
losses  in  the  convection  region,  103 
mechanical  equivalent  of,  15 

Historical  review  of  the  theories  of  cyclones  and  anticyclones: 
Bigelow's  asymmetric  cyclonic  system,  218 
Ferrel's  cold-center  anticyclone,  216 
Ferrel's  warm-center  cyclone,  216 
Hann's  dynamic  cyclone  and  anticyclone,  217 
Horizontal  magnetic  force,  semiannual  variations,  333-335 

I 
Inner  energy,  applications  of  formulas,  77 

bound  energy,  67 

formulas,  66 

in  mixtures,  29 

in  the  isothermal  region,  93-95 

in  the  planetary  circulation,  124 

kinetic,  26 

potential,  26 

total  heat  and  work,  26 
Integral  mean,  of  temperature,  36 

'1L37 
T 

of  pressures,  22 
lonization,  in  atmosphere,  292 

conduction  currents,  296 

electrostatic  relations,  295 

notation  and  formulas,  293 
Isobars,  observed  and  local  components,  224 
Isotherms,  normal  and  local,  235 

observed  and  local,  239 
Isothermal  region,  in  Europe,  91,  94 
in  the  Tropics,  95 

K 
Kinetic  energy,  of  motion,  67 

in  planetary  circulation,  120 
theory  of  gases,  for  the  atmosphere,  30 
constants  and  formulas,  31 
Huron,  Sept.  1,  1910,  410 
Uccle,  Nov.  9,  1911,  412 

L 

Land  cyclone,  observed  temperatures,  255 
observed  velocities,  255 


428  INDEX 

Latent  heats,  definitions,  69 

examples,  78-80 

second  form  of  the  equations,  75 
Leakage  currents,  Bigelow's,  219 
Lindenburg  observations  in  balloon  ascensions,  54-64 
Local  circulations,  general  equations  applied,  163 

M 

Magnetic  aperiodic  vectors  along  the  meridians,  329 

semiannual  inversions,  334 
field,  coordinates  and  forces,  360 
horizontal  force,  observed  variations,  330-331 
sphere,  theory,  363 

Margules'  paper,  on  the  theory  of  storms,  133 
Mass,  in  mixtures,  29 
Meteorological  elements,  annual  variations  in  Argentina,  338 

in  the  United  States,  338 
Meteorology,  constants  of  the  static,  dynamic  and  thermodynamic,  9-10 

status  of,  1 
Mixture  of  gases,  formulas,  23-29 

N 

Non-adiabatic  atmosphere,  application  of  formulas,  77 
general  formulas,  50 
working  equations,  53 

O 

Operator  V2,  in  rectangular,  cylindrical,  polar  co-ordinates,  146 

P 

Partial  formations,  in  the  annual  curves,  341 
Physics,  fundamental  laws  of,  28 
Polarization,  of  sunlight,  description,  348 
examples,  350 
percentage  observed,  351 
Potential  gradient,  definition,  144 
Precipitation,  annual  variations,  337 
Pressure,  compared  with  temperature  observations,  103 

dynamic  and  barometric  units,  112 

gradients,  rectangular,  cylindrical,  polar,  143 
evaluations,  157 

in  anticyclones  and  cyclones,  107,  108 

in  isothermal  region,  92-95 

in  mixtures  of  gases,  28 

semi-diurnal  waves,  100 

three  types  of  units,  P,  p,  B,  2-3 

variations  of,  17 


INDEX  429 


Problems,  general,  in  physics,  417 
Processes,  adiabatic,  68 

Carnot's  cyclic,  72 

cycle  for  vapors,  73 

irreversible  and  reversible,  70 

isenergetic,  68 

isodynamic,  66,  68 

isoelastic,  66,  69 

isopiestic,  66,  69 

Prominences  on  the  sun,  annual  variations,  337 
Pyrheliometer,  instrument,  262 

annual  mean  variations,  402 

example,  264 

first  method  of  discussion,  364 

reduction  to  sea  level,  399 

second  method  of  discussion,  395 

theory,  263 


Radiation,  calories  of,  288 

co-efficients  and  exponents,  82-90 

constituents,  375 

depletion  from  the  cirrus  maximum,  274 

determination  of  the  intensity,  265 

effective,  388 

energy,  in  the  planetary  circulation,  126 


formula,         -*,  290 

mean  observed  values,  291 

function,  K  =  c  T°,  1 
formulas,  82 

intensity  at  different  elevations,  271 

intensity  at  the  earth,  280 

in  the  isothermal  region,  86 

in  the  planetary  circulation,  124,  129 

maximum  value,  274 

mean  annual  values,  272 

relative  absorption  by  aqueous  vapor,  275-276 

selective  and  black,  388 

solar  and  terrestrial,  375 

total,  transmitted,  absorbed,  377 
Relative  humidity,  observed  in  the  atmosphere,  33 
Remarks,  general,  404 
Residuals,  339 

Rocky  Mountain  Plateau,  temperature  gradients,  34 
Rotating  earth,  equations  of  motion,  cylindrical,  148 
equations  of  motion,  polar,  149 


430  INDEX 


Solar  constant,  derived  from  the  bolometer,  291 

derived  from  the  pyrheliometer,  276 

derived  from  thermodynamics,  292 
physics,  magnetic  data  at  the  earth,  353 

siderial  and  synodic  periods,  354 

thermodynamic  equations,  352 

velocities  of  motions  in  sun-spots,  352 

velocity  of  rotation  in  latitude,  352 
Specific  heat,  at  constant  pressure,  1 

examples,  77-79 

in  isothermal  region,  93-95 

in  terms  of  latent  heat,  77 

variations,  38 

variations  in  monatomic  gases,  27 
Spherical  astronomy,  application  to  the  sun,  356 

formulas,  357 

positions  on  the  sun's  equator,  357 
Stefan's  radiation  formula,  279 
Stoke's  current  function,  Eq.  (494),  (495),  for  funnel-shaped  vortex,  172 

for  dumb-bell-shaped  vortex,  172 
Strata,  interflow  of  warm  and  cold,  248 
Summary,  394 
general,  403 
of  dimensions,  408 
Sun-spots,  annual  variations,  337 
Synchronism,  solar  and  terrestrial  phenomena,  338 
Synchronous  variations  of  the  elements,  335 


Tangential  angle  i,  in  circulations,  251 
Temperature,  annual  variations,  338 

first  distribution,  365 

gradients,  observed  in  the  atmosphere,  32 
computed  for  the  plateau,  34 

in  cyclones  and  anticyclones,  105,  237 

in  semi-diurnal  waves,  100 

in  the  isothermal  region,  92-95,  115 

in  the  planetary  circulation,  118 

of  the  sun,  probable  values,  281-285,  403 

second  distribution,  380 

standard,  3 

variations  of,  17,  37 

virtual,  37 

Theory,  change  of,  394 
Therm,  unit  of  heat,  16 
Thermodynamics,  first  law,  65 


INDEX  431 

Thermodynamics,  second  law,  70 

terms,  374 
Transmission  of  energy,  in  the  isothermal  and  convectional  region,  287-292 

U 

Units,  equivalent  values,  1 
of  heat  and  work,  14-1G 

systems  of:  (M.K.S.),  (C.G.S.),  (F.P.S.),  2,  408-409 
transformations  of,  12-14 


Variations  of  P,  p,  R,  T,  17 

Vector,  definitions  of  velocity  and  direction  of  motion,  7 

Velocities,  in  cyclones  and  anticyclones,  106-109 

in  different  latitudes  (velocity  function),  123 

in  the  planetary  circulation,  120 

linear  absolute  and  relative,  161 

normal  and  local  in  cyclones  and  anticyclones,  240 

normal  and  local  in  storms,  221 

observed  in  different  levels,  226 
Viriol,  of  inner  energy,  26 
Vortex  formulas,  dumb-bell-shaped  tube,  172 
funnel-shaped  tube,  172 
general  formulas,  174 
tubes,  successive,  173 
Vortices,  composition  of,  212 

connection  between  the  funnel  and  dumb-bell,  252 

cylindrical  equations,  168,  175 

evaluation  of  equations,  256-259 

examples  of  the  structure  in  the  atmosphere,  196 

examples  of  (1)    Cottage  City  waterspout,  199 

examples  of  (2)    St.  Louis  tornado,  202 

examples  of  (3)    DeWitte  typhoon,  205 

examples  of  (4)  Ocean  cyclone,  209 

general  equations,  213 

reversed  dumb-bell,  213 

W 

Westward  drift,  224-229 

Wien  displacement  formula,  280 

Wien-Planck  formula,  of  the  spectrum  energy,  278 

evaluations,  281 
Work,  energy,  formulas,  66 
applications,  81 

erg  as  unit  of,  15 

heat  equivalent  of,  16 

in  the  isothermal  region,  93-95 

in  the  planetary  circulation,  124 

total  work  done,  27 


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